Loot Systems

Someone invited you to a party and now you can't loot your own kill anymore !
Don't worry. You just got told that the "grouping is now changed to group loot". This means that the actual looting will be done in turns. Your turn will come.
Some quest items can be looted by all people (on that quest) at the same time, so you don't have to kill the same baddy a dozen times over.

There will be a cut-off (generally at 'green' or 'uncommon' items) where a Roll system pops up. There's a 'coin' (greed) and 'dice' (need). You can also pass using the [x] to close the pop up. How each group deals with this might vary, so if you're unsure, it's best to check up front with the rest of the party. (You can use /p to switch to the party chat. E.g. /p Hey Guys !)

If you hit Need, it will be for -that- character to use and equip. Not to sell it for profit (but I need that mount! is not a valid excuse), or to send it to an alt (everyone can roll a dozen alts). Generally, custom requires for your character to actually equip the item on the spot.
If you hit Greed, you just want to sell it in the Auction House, disenchant it (no, that's not a Need either) or send it to an alt.

First, the system will check if people have hit Need and roll for them. Highest roll wins it and that person receives the item. (Occasionally, people roll the same highest roll, in that case the game decides randomly on the lucky winner.) If noone needed the item, the people who pressed Greed get their turn.

Some items are Bind on Pickup (BoP). This means that you can't trade 'em to other people after you acquired it. This goes for Quest items/rewards, but also for drops you might get in an instance. Be especially careful with these, as people tend to get annoyed when you hit 'Need' on a BoP item, confirm that you want to 'Need' it... and then can't even(/ever) use it.
If noone wants needs a BoP item, it happens that the Enchanter in the party will 'Need' it and then Disenchant ('Shard') the item. People will then do a manual roll (/roll) to see who gets the result of the Disenchanting.

Another system you might find used in your party is the "Master Looter" system. This will set one person to be the Master Looter. Anything above the cut off level (again, generally green or better), will only be visible to them. They can then loot the corpse and decide who to give it to. Green BoE (Bind on Equip) items will generally get a random roll for the lucky winner, Blue (rare) items will get a check for who wants/needs them. On the one hand, this system is safer because there can be no 'mistakes' with BoP-needing. It's also faster than the regular group loot if you have a good Master Looter. It can be abused, obviously, so it's important to know what's going on and that you trust the Master Looter here.

Aggro

Fighting in Instances works differently from fighting outside by yourself. (Even fighting in groups is different from soloing). Monsters don't stop chasing you after a while, but follow you to the entrance of the instance. All running generally does is delay the group.

Monsters have a system for deciding who to attack. The way this works is called Aggro. Monsters decide who are the biggest threat to them and then go attack that person.

Simply coming close to a monster will lead to it attacking you. Dealing damage to a monster causes threat. Healing a party member causes threat. Some abilities reduce threat (e.g. Rogue's Feint) or increase it (e.g. Warrior's Sunder Armour or Heroic Strike). In short, you want the Monsters to focus on the Warrior and ignore the rest of the group. This makes your life easier, as the healer only has to focus on keeping one person alive, and Warriors are a lot easier to keep alive than, say, Mages.
Dealing too much damage on a Monster might make it focus on you instead. Having a few monsters and the Warrior only focusing on one will mean that the rest will likely go for the Healer instead. The first instances will probably see you trying to work this out properly, but it will become increasingly important as you progress through the game.

Raids

It's also possible to extend your group to be larger than 5 people. This is done through transforming it into a Raid group. (Hit 'o' to open your Social window. Then go to the 'Raid' tab. If you are the Group Leader, you can click 'Convert to Raid' to turn your regular group in to a Raid group). In a Raid group, you can not finish regular quests and you'll get a lot less experience per killed Monster. Some quests though have to be finished in a Raid group. The first Raid instance you will find is Blackrock Spire, which is accessible to a 10 man Raid. After that are the End Game instances. Zul'Gurub and Ruins of Ahn'Qiraj are for 20 people. The remaining Raid Instances are for 40 peope. Don't worry about this too much yet, as it will take a while for you to get there.
You are more likely to find yourself in a Raid Group when you hit the Battlegrounds (more on that below).

First instances on horde/alliance side

Horde: You get it easy, with the first instance smack-bang in the middle of main town Orgrimmar. RageFire Chasm (RFC) is aimed for characters around their 12th-16th level. You can find quests scattered all over the place (including at the other end of the world, in Undercity!). Be careful not to step into the lava =)

The Wailing Caverns in the Barrens are up next, those are aimed at characters around their 20th level.

Alliance: You'll have to wait till your 18th level or thereabouts to enter the Deadmines in Westfall. There will be quite an extensive quest line leading up to the final confrontation and it continues from there into the next instance, the Stockades in Stormwind (for mid 20s to 30 or so).

The Alliance can do quests in the Wailing Caverns, but getting there might be a problem on PvP servers. On top of that, the zones you'll be passing through may prove to be a bit deadly for level 20 characters.


Battlegrounds

Currently, there are three different Battlegrounds. Warsong Gulch (WSG), Arathi Basin (AB) and Alterac Valley (AV).
The level ranges for these are:
WSG: 10-19, 20-29 ... 50-59 and 60
AB: 20-29, 30-39 ... 50-59 and 60
AV: 51-60

In these, you'll fight a team of the opposing side. WSG is a classic capture the flag, AB will see you trying to control Resource Nodes and AV is a full out war where you attempt to kill the General of the other side.
WSG is for a 10 man raid group, AB for 15 and AV can hold up to 40 people on either end.

In WSG and AB you can enter as a (pre-made, or fixed) group, in AV you can only enter as a sole person.

Fighting and winning in these Battlegrounds will see you with Reputation and Honour. Honour ranks give you access to armour and weapons, and each Battleground will have rewards specific to them when you get high enough Reputation there. Exact details on these rewards can be found at the official site.

Particularly in the 10-19 and 20-29 bracket it's well possible to run into 'twinked' characters. These characters have the best gear available at that level and high level enchantments, turning them into very dangerous opponents. This practise gets more common the older the server gets.