This article discusses in depth how armor values effect damage reductions in Diablo 3. I play a barbarian and I think they can benefit from this information the most, although other classes can benefit just the same. Just to warn you, this article gets mathematically technical..

Diablo III Edit. As of Patch 2.1, Dexterity increases Armor (1 Armor for 1 Dexterity), in the same way Strength does for all classes. Monks and Demon Hunters (both use dexterity as their primary attribute) also increase all damage output by 1% for each point in Dexterity. Mar 03, 2020 Season 20 begins March 13, shortly after our next Diablo III patch. Read on to learn more about the brand-new Season buff, three brutally feral new class sets, and a handful of item reworks coming in Patch 2.6.8!

The graphs below aren't super exact because they don't take into account things like: Buffs, passive abilities, 30% DR for barbs (makes vit more powerful), block, increased mLvl in inferno. So don't live your life by them, but it's good for getting a better handle on how armor, resistance, and vitality work together.
I have witnessed many people explaining to others that armor has diminishing returns. While it is true that the amount of damage reduction goes down as armor goes up, the actual effect of armor on your survivability remains constant.
NOTE: If you don't feel like reading the technical math stuff, just jump down to the TLDNR section where there is a graph showing effective health vs armor.

Terms & Definitions:

Ancient items truly shine in Season 17: so long as you have no Set bonuses equipped, each Ancient you equip will provide you with a massive 750% boost to damage dealt and 4% reduction in damage taken. For those of you who may have taken a break from gathering loot recently, we thought a quick refresher was in order to clear up any confusion.

Armor:Armor gives you % damage reduction against all types of damage, not just physical. It is increased by armor value on items, strength (1:1) and certain skills.

Diminishing Returns: The idea that an extra point in something (Armor) yields less return than the points before it. Ex. the 100 point in armor is less effective than the 99th point.

Effective Health: Effective health is the amount of damage it takes to kill you (not to be confused with max health). If you have 1000 max health and 25% damage reduction, you have an effective health of 1333. You can determine your effective health by the following formula: MaxHealth/(1-DamageReduction)
This is where a lot of people get confused. Having 50% damage reduction does not allow you to take 50% more damage, it's a lot more than that. Consider this scenario: You have 1000 max health and 50% damage reduction. A monster is hitting you for 100 unmitigated damage, which only does 50 damage to you because you have 50% DR. It will take that monster 20 hits to kill you (1000 HP / 50 dmg) which means you can actually take 2,000 unmitigated damage (20 hits * 100 dmg). So with 50% DR you have 200% effective health, not 150.

mLvl: Stands for monster level. The level of the monster you're fighting.

DR: Damage reduced (from armor).
Explanation:
DR from Armor = Armor / (Armor+(50*mLvl))
As you can see, there is a constant in the bottom (50*mLvl). Armor is being divided by (Armor + 3000). It will always approach but never reach 1 because Armor can never be greater than or equal to (Armor + positive integer).
Graph illustrating damage reduced as armor increases:

The graph shows that as armor increases, additional points in armor yield less and less DR. Most people call this diminishing returns and explain that the more armor you have the less additional armor is worth. That is wrong. The reason it is wrong is because the effect of each additional point of DR is more powerful than the last. So in order to keep the effectiveness of armor linear, each point must must yield less damage reduction because each point of damage reduction is more effective than the last.
Graph illustrating your effective health as DR increases:

To further illustrate my last point lets look at an example:
-At 70% DR you have an effective health of 333.3%, increase over last = 10.8%
-At 71% DR you have an effective health of 344.8%, increase over last = 11.5%
-At 72% DR you have an effective health of 357.1%, increase over last = 12.3%
...
-At 95% DR you have an effective health of 2000%, increase over last = 333.35%
-At 96% DR you have an effective health of 2500%, increase over last = 500%
If every point in armor gave you the same amount of DR, each point of armor would be more effective than the least. Meaning that the best way to play the game would be to stack as much armor as possible. This is why Blizzard has balanced armor so that its relationship with effective health is linear. Most of you who have played WoW are probably familiar with this concept because they do it with almost all damage reduction stats. As a little side note, you may remember that resilience in WoW was slightly unbalanced so that each additional point in resilience was better than the last (but not by much).
Without further ado, here is the most important graph in the series. This graph shows your effective health as armor increases:

People should also note that the same thing holds true for resistances (and resistances via int). The formulas are basically the same.
DR from resistance = Resistance / (Resistance + (5 * mLvl))
Resistance from int = int * 0.1
Which boils down to, DR from int = int / (int + (50 * mLvL))
Which is the same formula as armor. So yes for int also the effect on survivability is linear.

TLDNR

Every 1000 points of Armor increases your effective health by 1/3 of your max health (with no other DR sources), regardless of how many points you do or do not have in Armor already. The same goes for resistances. Here is a graph showing effective health and DR as armor increases:

How this effects actual game play:
We all have the same question: Which is better for me? It depends on your current stats! (surprise, right?). For example, I play a barb with about 42k HP and 7000 armor. Currently 1 point of vitality is worth a little more than 8 points in armor.
-If Item One has 10 extra vitality and Item Two has 70 extra armor, I will go with Item One.
-If Item One has 10 extra vitality and Item Two has 90 extra armor, I will go with Item Two.
The same probably doesn't hold true for you. So I made it easy. Here is a dashboard you can use to input your current stats and have it tell you how good vitality is vs armor for you. You can also select 'Compare Items' and compare up to 5 items to each other based on their Vitality and Armor stats: http://www.swfcabin.com/open/1337704597
If there's any real use out of the thing I can incorporate +%life and resistances and stuff. But for now it just tells you Armor vs Vitality.

END TLDNR

Final Note on Resistance/Armor: There's a bit more to the whole thing that just that.
First, 1 resistance is basically always better than 1 armor, obviously. And 1 int is worth 0.1 resistance, so 10int > 1 armor.

Second, how good resistance/int is versus armor depends on your current armor and resistance. So don't necessarily stack one to the detriment of the other.

-The more Armor you have the better resistance becomes.
-The more resistance you have the better Armor becomes, relative to each other.

Now comes the question: which is better for me specifically? Well, it should be possible to tell how much armor is worth 1 resistance point. I have put together a matrix with Armor running across the top and Resistance running down the side, find where your armor and resistance intersect and it should tell you how many points of armor equal 1 point of resistance.
Resistance vs Armor Matrix
For me personally, my character's res/armor ratio is about 20. So if I have to choose between 500 armor and 20 resistance, I will choose the armor.
And to the people who are like 'omg what about int?' Well, I don't know how you got this far without realizing you can just take those values and divide them by 10, but here is a matrix showing the value of 1 additional point of int versus 1 additional point of armor.

What about Revenge?
Note: If you're interested in how this affects life leech builds, drop down to where it says LIFE LEECH.
Good question! I actually started looking into armor because I was thinking about revenge and how to get more use out of it. However, I quickly realized that revenge doesn't matter, and in fact no healing abilities do. Here's why:
Revenge heals you for 5% of your max hp. If you have more max hp, then you will be healed for more. Sounds like more vitality = better revenge, right? Wrong. If stack vitality, then revenge will give you a bigger heal. However, if you stack armor revenge will give you a smaller heal but those health points are effectively worth more because of your armor value. Either way though, it will heal you for exactly 5% of your max health which is also the same thing as healing you for 5% of your effective health. But anyways let me show you an example.
This might get hairy but follow along. Sorry I'm not the best at explaining things.
Lets say you have 10k HP and 1000 armor (25% DR). Revenge heals you for 500 (10,000 * 0.05). Which is worth 666.6 effective health (500 / (1-0.25) ). If we increase your vitality by 100, your health becomes 13,500 (10,000 = (100*35) ). Now revenge heals you for 675 hp (13,500 * 0.05), which is worth 900 effective health (675 / (1-0.25) ) when you have 1000 armor (25% DR). At those health and armor levels you can see that 1 point of vitality is worth about 14.1 points of armor. Now instead of increasing your vitality, we increase your armor by 1410 points (14.1 points for each increased point of vitality in the previous example). So you have 10k HP and 2410 Armor (44.6% DR). A revenge still heals you for 500 like in the beginning (10,000 * 0.05), but those 500 health points are worth 900 effective health points (500 / (1 - 0.446) ).
Here is a little worksheet showing that example.
So from this you can see that revenge heals you for 5% of your health, which gives you back 5% of your effective health. So actually all the ratio stuff between armor and vitality is completely unaffected by revenge and you should pay it no mind.
LIFE LEECH
Now what actually IS INTERESTING is the affect of other types of heals on the barb and how armor and vitality impact them, or rather don't. Any kind of heal that isn't based on a % of your max HP gets BETTER with more armor and is completely UNAFFECTED by vitality. This is because while it is healing the same amount, those health points are worth more when you have more armor.
Lets look at the passive skill bloodthirst. It is a barbarian passive that heals you for 3% of the damage you deal. Here is a spreadsheet showing the same example as above, but looking at the effective healing of 1 strike with bloodthirst active.
The moral of the story is this: If you want to try some kind of leeching build using bloodthirst and +life on hit mods, you should definitely stack armor and resistances over vitality to make your heals more effective. Also, I admit I don't have much knowledge of other classes but I assume they have some heals that are static or based on damage dealt. So anyone of any class trying to make a leech build should focus on res/armor over vitality.

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What does weapon items do for Wizards in Diablo 3? Does weapon item DPS matter? Do item qualities such as steal/+damage etc. transfer to actual spell damage? Can I use a regular attack? and more will be answered in this article!

Unlike in D2, a wizard should care deeply about his weapon in Diablo 3. It isn't just a “stat stick” with bonuses on it anymore. No longer are casters fairly equipment independent. All offensive skills have their damage based on the weapon item you have equipped. For example Magic Missile does 110% of your weapon damage, as arcane damage.
Weapon damage is used for spell damage. And weapon attack speed factors into your casting speed. Faster weapons will cast faster spells too.

How does Damage WORK?

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Base Damage
Lets take a look at Magic Missile, and a wand (weapon or axe) with a 8-12 damage range, once per second, this results in an average 10 DPS. This is what your weapon reports as DPS, the other stats are there, but in the small print.

The weapon damage is the actual damage the weapon has, not DPS. This is why 2 hand weapons are often favored. They do more damage, but slower. So they may do 15 damage, but only 0.65 times per second. That’s still 10 DPS. But the spell amplifies the damage more, as it starts with more.
+damage items, such as rings, increase this base damage. This is a key factor to increasing your total damage output, as it gets magnified by all other damage multipliers (spells, int, skills).
Intelligence Bonus
The intelligence percentage works like this. Each point of intelligence boosts damage by 1%. So 110 int, increases damage by 110%. That is a 2.1 multiplier (shifting it from % to decimals, and treating it as a 100% increase).
Spell Bonus
Magic Missile then comes into effect, and deals damage specified it by its spell effect, listed as 110% under the skill. So it takes the damage listed in your inventory screen, and does 110%. If you did 100 damage, MM will hit for 110 damage.
Putting it all together
To determine the spell damage you take your:
(Weapon Damage with +damage from items) * Intelligence % * Spell Effect
So, if we start with a wand that averages 10 damage we get:
Weapon Damage (10) * Intelligence (2.10) * Spell Effect (1.10) for: 23.1 damage.
Now, if you increase the weapon damage, the effect grows. Not only does the weapon provide a bigger base damage to start with, but the % increase from intelligence and the spell have a bigger impact. 10% of 15, is larger than 10% of 10 for example.
So if we use a weapon that does 15 damage (the extra 5 damage from the weapon, or +damage items) we get: 15*2.1*1.1 = ~34.6 Damage.
That 5 additional damage gave us a head start, but the int and spell effect also gave us ~19.6 damage from intelligence and spell effect, while the 10 damage weapon only gained us 13.1 from our base.
This means, especially at low levels, that an increase in weapon damage, either from the weapon, or +damage rings etc, can have a huge effect on your damage output. That +damage is added to your base weapon damage, so it’s also amplified by your intelligence and your spell effects. If you have to choose between +damage or +int, go for the damage.
At higher levels, when adding 5 or 6 damage to your base isn’t that big an increase the need to prefer damage over intelligence boosts evens out a lot more.
So where do I see spell DPS?
The short version of this is that the damage reported on your inventory screen is the damage the skills boost when you cast them. If you get that number up, through whatever combination of items, your spell damage will go up. It factors in all items and modifiers before the spell automatically.
The damage increase listed in the stat summary of the items, when you compare them, is accurate, as best as I can determine. If it says your damage will increase 10 points by swapping the items, it appears accurate when I test by monitoring Hydra damage.
There is only one catch. For the time being, when comparing a 2H weapon to a 1H weapon, the game only reports the damage increase from swapping the weapons only. It does not factor in the loss of any bonuses from your offhand item. After you swap out a 1H item for a 2H, check the inventory screen for the final result, listed on the left side of the inventory window. When comparing a 1h to a 1h or a 2h to a 2h, the item reports the correct change in stats.
Elemental or Bonus Damage on a weapon:
So you have a 10dps sword, with +2 cold damage…why don’t you strike for 12 damage?
--The damage listed on the weapon is already integrated into the weapons base damage. The DPS takes it into account. You can confirm this by socketing a Ruby into your weapon. The base damage jumps, AND it lists the rubies bonus…so it does the math for you, but lets you know in case you have any +Cold damage % skills, which will only effect the +2 cold damage (not the entire weapon damage).
--The spells you cast usually replace the elemental damage type with their own. +2 cold damage from a physical blow will chill or freeze an enemy. From Magic missile, it’s converted to arcane, and will not chill your target. The damage is still there, in the base damage. It just isn’t cold damage anymore. Some spells retain the damage type. Spectral blades IIRC will deal cold damage if using a cold weapon.
2H vs 1H weapons
Now, does this mean that bigger base damage (i.e. 2H weapons) are always better than 1H? First, that’s gear dependent. If you get a nice off hand item you may end up doing more damage if you factor in it’s bonuses. Second you give up any other perks the off hand item provides. So no blocking from shields, less magical abilities (only one items worth in the 2H, vs the two items worth in the 1h & offhand). That can factor into decisions.
Weapon Speed
Wizards still care about this, as many spells casting time is linked to the weapon speed. If you have a faster weapon you can cast more spells per second. This is really good if you rely on signature (free) spells, or have abilities that you want to trigger. This, btw, is often refered to as a 'proc' for programmed random occurance. For example the lightning rune on magic weapon can trigger extra lightning strikes. Faster casts mean more chances for it to proc. (Thanks Leetnoob)
However if you like big burst damage spells (arcane orb) you may just want more damage. The limiting factor for them is not how fast you can cast them, but that you can only cast 3 or so before you are out of AP. Some spells, like Hydra, are extreme examples. You only get 1 hydra….doesn’t matter how fast you cast it, so more damage outweighs faster weapons, so 2H weapons win there. (Thanks Morphos)
For channeled spells (rays and torrent) the 2H weapons can still out perform faster 1H weapons, and this comes down, again, to mana cost. The two weapon types will have similar DPS, but the 1H weapons trigger more often (as seen by a 'pulse' in the beams), draining AP more often. So you spend more AP per damage point, for the same DPS. Unless you have some abilities you want to proc, this may be to inefficient for some builds & gear sets. (Thanks Dedna & JumpSec)
This doesn't mean 1H weapons (with a good offhand) are automatically worse. It is possible to make up for the AP 'inefficiency' of 1H weapons. Faster casting essentially costs more AP per second, meaning a fast caster will be out of AP and have to wait for the recharge. The advantage of the 1H is that they won't just wait during this phase. A skilled player will spam the 'free' signature spells during this time. This may be enough to close the damage gap between the weak AP spells of faster casts, compared to the strong AP spells of slow casts.
It can also be argued that it allows more flexibility and control. A fast caster will throw out 3 weaker Arcane Orbs (AO) really fast to smash the front ranks of a crowd, then fire off some chain lightning to catch stragglers, or try to get the back ranks (where summoners like to hide). If you throw in some other abilities triggered on each hit (paralysis, AP on crit) you may make up for the AP inefficiency in other ways (signature spells runed for AP gain)
Whether or not it's worth the extra work is up to the user, the available gear and their playstyle.
Damage Boosting Skills (MAGIC WEAPON)
Magic Weapon boosts your weapon damage, which increases spell damage. This bonus is applied alongside the intelligence and spell effect boosts. So a 10% boost from magic weapon on a 100 damage item, makes it a 110 damage item. If you have a 200% intelligence boost, and spell effect (to make the math easy) that gives you 440 damage output. Without the magic weapon, you would get 400 damage.
Note: Magic weapon may be bugged, and providing a 20% damage boost instead of the 10% reported on the skill.
These skills all stack, so if you get 10% from MW, 20% from the slow time rune, those both multiply your base damage. So the 100 damage item is run through:
100 damage * 2 (200% intelligence) * 2(spell effect %) * 1.1 (10% MW) *1.2 (slow time rune) = 528 damage
Some skills, like Familiar, do not boost the damage, as they fire their own bolt of energy at your target. The end effect is the enemy takes more damage (at the 20% familiar reports) it’s just not from your arcane orb, or other spells. That is unless you select a specific rune.

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Quick FAQ to sum all this up:
1) Do Weapons MATTER for a wizard?
Yes. All damage dealing skills are based off of weapon damage.
Want to know how much your weapon damage is boosted by spell?
--Hold 'ctrl' while hovering over the skill.
--Or select 'advanced tooltips' from the 'gameplay' option menu.
But wizards shouldn’t use axes!
If you say that doesn't make sense...I say it's MAGIC!
If you think it makes wands/staffs obsolete…they get comparable DPS to other weapons (compare wands to 1h ranged, and staffs to 2h weapons), and get wizard desired bonuses (+AP on crit, etc).
2) Want more spell damage?
In general order of effectiveness:
--Equip a weapon that has a higher base damage.
--Equip items with +damage bonus (this is added to base weapon damage).
--Equip items with +intelligence bonus (this multiplies base weapon damage).
--Use skills that boost weapon DPS. (Magic Weapon). Boosting weapon damage, boosts spell damage.
--Higher attack speeds affect cast rate…this one is a bit more complicated, but all other things being equal, higher attack rates win.
3) Do Weapon effects transfer to spells?
Yes. Life steal, experience gain, damage, it all transfers to your spells.
4) Where is the regular attack? I want to shoot my bow!
Your signature spells (your 'primary' skills) are free to cast, and will do more damage than your base attack, plus other nifty status effects. There is no real reason I can imagine to want your basic attack. Other classes don't use their basic attacks either, and even in D2 it usually fell to the wayside.
If you want it: Merely drag your selected skill off of the primary or secondary skill slot. The default is a normal weapon attack.
If it's because you want a close range spell, like spectral blades, and a long range, like magic missle at the same time, but can't set it up...you can!
5) Want two defense spells? Two signature spells?
Turn on 'Elective Mode' in the gameplay options menu and you can now map any skill to any hotbar slot. So you can have both slow time and diamond skin, or spectral blades and magic missile.

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For more in-depth guides for Wizards I would check out the Diablo 3 Gold Secrets By Peng Joon guide.