Know your role
When you queue for a random dungeon, the tool will ask you to identify which role you want to perform. The three roles are tank, healer, and damage dealer (DPS, or damage per second). Tanks are tough, damage-resistant, and good at keeping the enemy's attention. Healers are able to buff, clear debuffs, and restore the hit points of party members. DPSers do a lot of damage, a necessary component in having the enemy die first.
Of course, Cataclysm has reintroduced an important aspect to many classes that had been previously been deprecated. Many classes are capable of performing crowd control. These are abilities like the mage's Polymorph and a retribution paladin's Repentance. These crowd control abilities force enemies into a time-out and not get involved in the fight.
In Wrath of the Lich King, tanks could pull entire rooms and stand there, using area effect abilities to keep all the mobs focused correctly. That's no longer possible. Now, if an entire large group of enemies swarm the tank, the healer is unlikely to have the mana to keep up with fight, and the tank is unlikely to hold aggro. You should get comfortable with the crowd control abilities of your class.
Let the tank do the tanking
One of the tough parts in making the transition from world questing to instances is what's known as "the pull." The pull is the first shot at the bosses, the first few attacks. During the pull, the tank needs to get mobs lined up, get their attention focused on him, and make sure he's got control of the situation. Sometimes during the pull, the group will decide to use crowd control as part of the initial salvo, but that's something you should work out before doing the pull.
The important thing here is to give the tank time to do that task. Even if you're a class that wears plate armor, if you're in the damage-dealing role, you're probably not equipped to take much of a beating. If you distract mobs away from the tank at any time, the tank will have a much more difficult time getting things under control. Give the tank a bit of time to get the pull finished and aggro well-established before opening up the cannons. If you're not sure how long to wait, 5 to 10 seconds is a good rule of thumb.
Follow a kill order
When you're fighting a large group of mobs, most groups will establish a kill order. This is the order in which you burn down the mobs. If you use the raid markers included in the games interface, most groups go in order of skull, red X, and then blue square. If in doubt, usually, kill spellcasting mobs first, then melee mobs.
The important thing, however, is to use an assist command. You can do this by targeting your tank and pressing "F" on your keyboard. By using this technique, you'll be sure that you are attacking mobs the tank is attacking, and you won't risk pulling aggro off other mobs.
Don't stand in that
The rule to "don't stand in fire" has almost become a joke at this point, which is a shame. Not standing in stuff is one of the most important skills you can have while performing dungeons. Bosses and enemies breath fire, summon black spots, and otherwise create a lot of environmental hazards. Your most important job is to simply not to stand in that stuff.
This is important because if you do stand in fire, your healer will have to exert extra focus in keeping you alive. That costs mana, focus, and cooldowns that the healer may need to use for other unavoidable damage. While you'll occasionally take damage from environmental hazards faster than you can possibly react, you should still try and keep from taking more damage than necessary.
Take the time to say hi
It's kind of annoying to join a group that's cold, silent and completely untalkative. While you tend to join dungeon groups in order to kill mobs, complete quests, and earn gear, that should still be an enjoyable process. It's hard to feel like you're having fun with new comrades when no one can be bothered to say "hello."
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.