LFG stands for Looking for Group and eventually, you're going to need a group. It might be for a group quest or, more likely, to run a dungeon. There are a couple of ways to find a group and some things to know before you head out on your group adventure. There's a built in LFG feature within WoW pictured to the left. This puts you in a list that people can see and then they might invite you to join a group.
While this is sometimes effective and I like to use it if I'd like to join a group, but not immediately, if you really want to do something with a group, it's often better to just ask in Trade or General chat. In a major city, trade chat is usually more active, but if you're out in the world, general chat is going to be your best bet. To find a group this way, simply type /trade (to get you into trade chat; use general if you want to chat there) and then type something like "LFG ToC." Now, you have no idea what that means. Most people use acronyms for all the dungeons and you'll look less like a noob if you do too. The acronyms can be found at sites like WoWwiki and WoWHead. It's sometimes also helpful to let people know what role you can play. So, you can say "DPS LFG ToC" which tells people that you're a damage person looking for a group for the Trial of Champions dungeon.
Another way to find a group is simply to monitor chat and when someone else announces that they're looking for a group or looking for a particular role for a dungeon or quest you're interested in, you can ask to join. Of course, if you're in a guild, you can ask your guildies to join you for an adventure.
Lingo alert! A group formed spontaneously is called a PUG or pickup group. Think of it like pickup basketball.
Once you're in a group, it helps to figure out what the expectations are. Group chat is /party and that's where you can start these conversations. You might want to clarify roles--who's tanking, who's healing, etc. Most importantly, you'll want to clarify loot rules, especially as you get into higher level dungeons where the loot is awesome. Any time something drops from a kill that is green, blue or purple while in a group, you get a dialog box (right). You roll on these items by clicking the dice or the money icon. You can pass by clicking the x to close the window. The dice is a need roll. Need rolls trump greed rolls (money icon), and are usually only used if you really want something. Most of the time, people roll greed on an item or pass on items that they really don't need. You might ask at the beginning of a dungeon run whether most stuff is need or greed. If something drops that you really want, don't be afraid to ask if you can need roll. I always ask before I need and so do most people I've been in groups with since need rolls trump greed and some items become non-tradeable once you pick them up. This has changed a little bit and even bind on pick up items are often tradeable to other players in the group for a certain period of time. Still, it's always good to ask. There's nothing more annoying than someone needing something that you really wanted. More than one person can need roll on an item and if two or more people really want something, that's generally the way it's handled.
If you've never been to a dungeon before, don't hesitate to ask for advice about what to do. Most parties end up with one or more people who've run a dungeon several times and they're more than willing to tell you what to expect during a boss fight or more generally. They'd much rather explain it to you than have your lack of knowledge cause everyone to die. Often, the tank will mark mobs in the dungeon, having everyone focus on one mob at a time. It's a good idea to follow the order, again, so that everyone won't die.
Group dynamics can get especially weird if things don't go well. Personally, I've never had the experience of being the new person causing everything to go bad. Generally what I see happen is that someone in the group isn't geared enough or is being haphazard about their play style (like not waiting until the healer has enough mana to heal or jumping into a fight when not everyone is ready) and causing everyone to die. If you're lucky, the person will realize that and will quit voluntarily. Sadly, more often, I've seen people blame everyone but themselves and then quit in disgust. Good times. And sometimes, you just have a bad combination of people. You might really need a ranged dps to win a fight and all you have are melee. It's okay to say, hey, this is working, I think we should call it. Because when you die a lot, you're going to have to repair, and repairing costs money.
Group experiences are really fun, though, and it's a great feeling to work together to beat a really complicated boss or make it through a long dungeon. Now that I'm at max level, I find group experiences, whether with my guildies or with a PUG, to be my favorite part of the game.