Choose your class
Okay, it might seem needless to say, but if you're going to tank in WoW, you have to choose the right class. Your options are to be a death knight, druid, paladin, or warrior. Paladins and warriors use a shield to help mitigate large portions of damage, while death knights and druids use a combination of raw gusto and extra abilities to do the same.
While each class has its benefits and drawbacks, the tanking class you choose is largely a matter of personal choice. To reduce them to soundbites can be difficult. In general, paladins have a large defensive toolbox, death knights have unparallelled control over mob placement, druids are made of raw stamina and mitigation, while warriors mix aspects of all four.
Setting the pace
The tank's job is made up of two primary roles. You take and mitigate damage, and you keep the mobs from attacking other party members. Really, everything else you do boils down to those two things. An important part of controlling mobs' attacks is "the pull." You set the pace for that. It doesn't mean you're necessarily leading the dungeon, but you give the "Okay, go!" for the time to pull.
If your group is using a lot of crowd control, you generally want to let that CC happen and then grab the mobs "on the inbound." That means letting the mage Polymorph the target and then picking up the rest of the pack before they turn the mage into a fine, arcane-flavored jelly. You will need to be ready with your biggest multi-target attack as the rest of the mobs get into range, or you might risk losing a target or two.
If you're not using any crowd control, you need to pull first. That initial "face aggro" isn't really a big deal, but it does give you the choice about how to control the mobs.
Framing the pull
There are a couple of different ways you can execute a pull. A lot of pulls are what I call the "pick a fight" method.
When you pick a fight, you're just charging into the middle of a pack and firing off your threat rotation. Many tanks like to use a distance attack and wait for them to come nearby. I don't do that because, invariably, DPS likes to attack the mobs before they ever reach me. If I need to create distance between those mobs and my location, I'll at least meet them halfway.
The other common method is the "LOS" pull. In this case, you use a ranged attack to the get the mobs' attention; then you go around a corner. Since the mobs have to come around the same corner to cast spells at you, they follow and end up nearby. The "LOS" pull is a time-tested way to make the mobs come close to you.
Picking your target
A tank chooses the kill order. Even if someone else is the dungeon leader and that person is marking targets, the tank still ratifies that kill order by issuing their threat on mobs (which in turn makes it safe for DPS to attack that mob).
It really really helps to have at least the Skull mark and X mark available as keybinds or macros. That way, when you switch onto a target and start opening a can of whup-butt on it, the DPS knows that mob is safe to attack. You just slap your keybind to tell them which mob to attack. You can set up that keybind under Key Bindings.
As for who you should actually attack first ... that depends. A lot of dungeons have special mobs that do special things, and you will need to learn which is which. As a rule of thumb, though, kill the one in the dress. If there's a spellcasting mob, kill that one. Spellcasters just hit so much harder than melee, in general, and do wild things like sheeping party members.
Gearing your tank
Your ability to mitigate damage is your other main job. While you do a lot of this job with cooldowns, damage mitigation is largely performed by your gear. There was a time when you had to worry about defense caps, complicated avoidance numbers, and things like that. No more!
Essentially, a tank needs to worry about mastery, dodge, parry, and block. Stamina comes with the item level of your gear, so take the highest item level gear available. Warriors and paladins love mastery; mastery helps ensure they block every incoming attack. Druids and death knights are a little more interested in the armor and dodge. Death knights can also parry, but druids do not parry.
Gearing comes quickly for tanks as they work through normal dungeons, heading toward heroic dungeons. The problem is that you have to start somewhere. Start in normal dungeons; it's the best you can do.
Stick with it and don't stress
As you're beginning your journey as a tank, there will be periods when it feels particularly painful. After a few wipes, you might feel like you're far too squishy to tank anything. Don't worry; you're just learning. Just keep with it, do your best, and the skills will come.
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.