General DPS Primer

The Dps'ers job
is to:
1) not get hurt if you can avoid it
2) not pull aggro (this is really just the most important thing in achieving 1)
3) do as much damage to the mobs as possible

Those priorities are ordered for a reason:
You might be able to get away with ignoring 1 and 2 if your healer and tank overgear you enough, or are good enough to pull your ass out of the fire. However that will not make you any friends and getting into that habit pretty much guarantees a wipefest in instances and raids that are at the limit of your gearlevel (lot's of people doing so is also the main reason why most tanks and healers despise pugging).

Keeping track of how you're doing
There's 2 addons you'll want to install:
1) Recount: While often billed as just a dps meter, it's real value lies in allowing you to figure out which abilities improve your dps the most, and where your problem areas are. See this article for a tutorial to using recount

2) Omen Threat Meter: this is your basic threat meter. The reason a dps wants this is because it will show you when to slow down a bit to avoid pulling aggro on a mob (especially important when you either seriously outgear the tank, or the tank is struggling for whatever reason)

General Applicable Tips to avoid damage/stay alive
1) Wait for the mobs to cluster and the tank to get aggro before you open up (if you find the tank is having trouble give him an extra couple of seconds at each pull before barging in and opening up). NOTE: this means you let the tank pull unless asked.

2) if you're not sure the tank has solid aggro on all the mobs don't use AoE abilities (in general it's usually save to AoE after skull-mob is down)

3) don't (keep) stand(ing) in poison spots (green or purple usually), black disks, lava pools, or any other kind of environmental damage. NOTE: avoiding damage is more important then doing ?? more dps over the course of the fight so this gets priority over getting another hit in.

4a) If you're melee always try to position yourself to the side of the mobs as this ensures you won't be hit by any cone-style damaging abilites of the mob, and most bigger mobs/bosses have at least 1 cone-style attack (being at the back of the mob is the alternative, though for mobs with a big tail that's often a spot where you get damaged by a tail-attack so aim for the side when possible).

4b) if you're ranged you usually want to be at maximum range and spread out, except on fights where the boss charges (e.g. rhino in hall of trophies in UP), or where adds will appear and likely need to be pulled of you (e.g. first boss in UK) in which case you want to be just outside of melee range (so tank can aoe pull the adds, or in the UP example so that charge distance and thus damage is minimal)

5) follow the marked kill order (skull, cross, square, moon) as it decreases the chance you'll pull aggro, and if the tank knows what he's doing the mobs are marked in a certain order for a reason. If mobs are unmarked then generally mobs that fear are priority 1 targets, ranged and casters mobs are priority 2 targets, and mobs that immobilize (stun/root/freeze/daze/...) are priority 3, after that you generally take the mobs with least health first.

6) if you get aggro drag the mob over to the tank, it makes it much easier for the tank to get aggro back. NOTE: if there are loose mobs and you pick them up the same applies.
To reinforce this point please watch the following >youtube song< till it starts playing through your head whenever you get aggro

7) make sure you position yourself in healing range (so not out of line of sight and not to far)

8 ) Know the de-buffs/stuns/... that your class has and use them on mobs when that's applicable to reduce incoming damage (the addon debuffEnough is handy to remind you of what debuffs to cast)

9) Do use self-healing abilities/bandages/potions when possible, the healer's priority is keeping the tank up, if the healer is hard-pressed you will be left to fend for yourself.

General Applicable Tips to increase the damage you do
1) don't die (as a dead player does not do damage)

2) make sure you're hit-capped (at the very least soft-capped so your yellow attacks don't miss, for dual wielding hard-cap is probably not reachable)

3) make sure you're soft-expertise capped so your attacks can't be dodged

4) if you're melee always try to position yourself to the side of the mobs as this ensures your attacks won't be parried (being at the back of the mob is the alternative, though for mobs with a big tail that's often a spot where you get damaged by a tail-attack so aim for the side when possible)

5) know your class+spec rotation. There's a lot of work done by various serious theorycrafters in figuring out the optimal ability use for each class. Make use of that work by looking up what the optimal rotation of your class is according to the theorycrafters and why (same goes for talent and glyph choices)

6) know the buffs of your class and use them for both yourself and party members when possible

7) make sure all enchant/gem slots are used (if you're still gearing up a low-cost gem/enchant of a couple of gold in AH is still a lot better then not enchanting or gemming at all)

8 ) know which stats are important for your class and spec, purple items are not necessarily better then blue, and higher iLevel isn't always better (for instance the 80's frost resist gear is iLevel 213 but a big part of that iLevel is spent on frost resist, so usually a iLevel 200 item is actually better outside of a frost resist fight).

9) if you're a new 80 do check out the blue crafted starter set for your class, it's almost certainly better then the green/blues/purple you've had since 70 or 72

10) get the consumables that'll will boost you (food/flasks/elixirs/scrolls). End-level buff food in particular can give you a pretty big boost and is cheap and easy to get (doubly so if you've leveled your cooking to 420 or so)

Oooh - that song is awesome