Are people going for armor, resists, vitality, damage? Which stat(s) and why? Does it vary with each class?
There are two ways to roll.
Either Tank/normal, or Glass Cannon.
In normal/tank mode, you want to stack both armour and resists (and preferably block chance if you're using a shield). You can use this handy guide to figure out how much armour 1 resist is worth at your current armour/resist levels. The reason people are stacking armour/resists and not just straight vitality is because of how damage reduction (DR) works. It works by increasing your effective hp (EHP). This means at 50% total DR, if you have 10k health, it's actually like you have 20k health, because a mob will have to do 20k unreduced damage to kill you.
Another benefit to having low actual hp, but high EHP, is that any heals are worth a lot more. If I have 100k health and no DR, and I heal for something that heals for 10k, I'm healing 1/10th of my total health. If I have 20k health, and 80% DR, even though I have the same EHP (100k), my heals are now worth half of my actual health, increasing the worth of heals (this doesn't work for any % based heals).
After DR, and some vitality, you want to stack enough dps stats to let you kill enemies before they enrage. These dps stats can be your main stat/crit%/crit damage/etc.
A few extra stats that help here (especially for Barb/Monk) are Life on Hit and Attack Speed %. Gear with these two stats can be worth a lot if they're a high amount, as attack speed % increases the effectiveness of Life on Hit, and Life on hit in high amounts (coupled with high EHP and lower actual HP) can be the difference between farming a difficulty and barely progressing.
For Glass Cannon mode you want to forgo any survivability stats (armour/resists/vit) and just stack DPS stats. This only works for the ranged classes (Wiz/WD/DH) because melee classes have to get close to fight, so they'll always get hit at least once.
This build works by having so much dps that you kill enemies before they can even hurt you. You also need to bring at least one survivability spell, because when you fight ranged enemies it can sometimes be tough to avoid their moves.
In this playstyle, all you need to do is shoot anything before it can get close. If it hasn't died in time, pop your survivability move and move away before shooting it again. You will be one shot by pretty much anything in Inferno with this build, but it's very easy to gear up for, so that's why people use it a lot, as opposed to the normal builds which cost a lot of money to gear up for.
You want to have significant values for all the stats you mentioned, and then more.
As an example, solo'ing Belial on Act II is extremely difficult, I have yet to beat him. Here are my stats:
I can't speak for any other class, but the above stats costs millions of gold to produce, and it still isn't enough for the act bosses. Although I can easily survive anything else in Act II, heavy hitters are still common.
It definitely varies with each class. For instance, Demon Hunters pretty much don't even try to build survivability stats, whereas with other classes survivability stats are pretty much a must-have.
In general, though, you're going to want a significant amount of your class' primary stat (e.g. Int for Wizards), then some mix of secondary DPS stats (e.g. Attack Speed, Crit, CritDamage), and some mix of survivability (Vitality, Resists, Armor, etc).
Resist all has been referred to as the "God stat," due to its necessity for barbarians and monks, as well as its usefulness to each and every class.
While it does vary for each class, the current stat preferences is quite helpful:
Resists > Vitality > Primary Stat (Strength/Dexterity/Intelligence)
My monk is currently able to solo Act II Inferno, here are my stats:
Resists to All: 71.34% Dodge Chance: 49.73% Block Chance: 16.00% Armor: 6831 Health: ~31,000 DPS: ~14k Attack Speed: 1.94 Life on Hit: 491
This is with me stacking 4 defensive skills and 1 offensive skill, Fists of Thunder and 2 heals + 2 immunities respectively.
If you're using a monk, comparative stats should be good enough to get you through Inferno.