Guilds (not clans) - Groups of people that can chat to each other, earn achievements together, raid together, socialize, run events, etc. It's mostly social. I recommending finding a friendly, social guild with people that are willing to help you, since most of the community are horrible to noobs.
Realms - Servers. There are 5 different kinds of servers;
PVP (Player vs Player - Members of the opposite faction can attack you without warning in most areas. Can be fun for more experienced players, but new players often get killed repeatedly for laughs)
PVE (Player vs Environment - Members of the opposite faction cannot attack you unless you set yourself as hostile. This is good for new players as they can quest and explore in peace)
RP (Roleplay - This is a server where most people don't really care about Raiding (PVE) or PVP. They pretend to be their character and interact with other people who are doing the same. This is great if you enjoy creating stories and socializing with people in a fun, creative way)
RPPVP (Roleplaying Player Vs Player - Same as a RP realm and a PVP realm - You can fight members of the opposite faction and roleplay while you do it :) )
New Players - This is a realm with people who are willing to help you learn to get to grips with the game. There is rarely any raiding or PVP here, as most of the players are noobs.
Dungeon - This is an instanced area (only your group of people - your Party - will be seen in the area) where you fight monsters to get through the dungeons and kill the bosses for gear, experience, quests, points and achievements.
Raid - This is an instanced area where you fight monsters (trash) until you get to one of the Bosses. These are like Dungeon bosses, but MUCH tougher. These take a long time to kill and require a lot of strategical gameplay and communication. You can enter a simple raid (Looking For Raid - LFR) when you are level 90 that you can just go in and hit the boss without doing too much thinking and you will get gold and gear for it.
Battleground - This is an instanced area that you fight other players of the opposite faction. These are 10vs10, 15vs15 or 40vs40. You must complete objectives and defeat the other side to win. Objectives are simply either Capture the flag from the enemy base and bring it back to yours without dying, Capture enemy bases, Deplete enemy resources by killing the enemy players or Kill the enemy's bosses (40vs40 only)
Rated Battleground (RBG) - Same as battleground but you need a team of experienced players. These require a lot of communication and strategy.
Arena - Basically a small Battleground where the only objective is to kill the enemy team. These can be either 2vs2, 3vs3 or 5vs5.