The game does provide a basic test to see whether or not you're eligible for instances. It checks your level before you can enter, and you're required to meet a basic gear item level to use the random dungeon finder. But we've all seen that you can game those systems in any number of ways.
For normal level 85 instances, DPSers want to be putting out somewhere between 5k to 7k DPS. That's a rough estimate; some fights favor certain roles, other fights favor certain specs. It's really a mixed bag, and you can't just assume that your DPS on a target dummy is "enough." Still, 5k to 7k DPS is a solid staple.
Tank and healer thresholds are a little bit more of a challenge to pin down, because crowd control is key in normal instances. Tanks should be in predominately 300 item level gear, as should healers. The closer your item level is to 333, however, the better off you are.
Assuming you don't totally outgear the instance, you'll want to use a lot of crowd control. The more successful you are in your efforts to use CC, the less pressure is on the tanks and healers. The gauge swings too wildly based on that CC to be any more specific.
Ready for heroics
Heroic instances are more demanding. Your DPS output should be somewhere between 7k and 10k DPS. You can squeak by with a little less DPS, certainly, but the 7k to 10k range is a nice, comfortable place to be.
Tanks should be rocking close to 100k hit points natively through gear. If you are geared to 100k hit points through item level alone, you should have no problem at all tanking heroics. Using enchants and gems to get you there, however, will mean you're largely a mana sponge and will thus be putting a lot more pressure on the healer.
Healers are the hardest to gauge, as always. A lot of a healer's performance relies on his or her ability to choose heals wisely. But your item level should be above 329, of course, and moving closer to the 340s.
Time for raiding
DPS characters joining raids should be putting out 10k or more damage per second. This is an easy gauge, since that's what's required to take down Argaloth
. Anything less than 10k DPS, and your raid will need to make up the difference. While other raid bosses don't necessarily have an enrage timer, Argaloth is a good, solid test of your damage.
Tanks should be over 120k hit points from their gear at this point. Healers should be at least 340 item level; everything over 346 is bonus.
The thresholds above represent rough guidelines. Please don't forbid a skilled player from your raid and quote this article as being why. However, for folks who are trying to get a good feel of whether they stack up in terms of forward progression, hopefully these goals will give you a deeper understanding.
Also, it helps if you're in a guild group. This is because the coordination and team play tends to be a little stronger, making up a bit for gear issues. Also, if more than one person in your group is new to that level of play, then you will be even a little more challenged in the learning curve. If you find yourself in a PUG or with more newer players, pad the numbers even a little further.
My favorite mod for gauging my DPS is Skada
. It's pretty solid and reliable, and I like its options.
Visit the WoW Rookie Guide for links to everything you need to get started as a new player, from how to control your character and camera angles when you're just starting out, to pulling together enough cash for mid-level expenses such as mounts, to dungeoneering and travel tips for lowbies.