Realms are the game servers characters are stored on. Choosing the right realm is important as this will change your gameplay immensely depending of the type of realm you pick.
Choosing a Good Realm
Most of the time, especially when you are new to the game you might want to choose a not so highly populated realm because it will make it easier for you to get the hang of the game and play in a relaxed environment with a community that's in the process of being built. When choosing a realm make sure to pick a realm which is marked as Recommended or is known to have Low Population. New realms are often opened, so keep an eye for updates in the realm list.
PvP means Player versus Player. In those realms the gameplay is concentrated on battles between the two factions (Alliance and Horde) and often times during quests you might get ambushed and killed by members of the opposing faction. This can be very frustrating, so if you cannot deal with it, don't roll a character on such a realm. I have experiences with a player who started on Skullcrusher (PvP) got to level 43 and one time when he was repeatedly "ganked" by some other players, he quit his character and deleted it because he was pissed.
There is a good side of playing in a PvP realm though. You will be able to gain honor much faster which means you will go up the ranks easily (if the realm is not overpopulated) and get honor rewards faster.
PvP realms have special zones for each territory you are on. There are Contested Zones and enemy zones (Alliance or Horde) where PvP mode is enabled automatically and you will be open to enemy attacks at any moment. The Majority of zones in the game are contested so it's going to be tough surviving sometimes while trying to do quests.
Click here for a list of zones and their respective types plus level requirement for questing/grinding.
Player Versus Environment realms are for those who enjoy questing and fighting computer controlled enemies rather than battling players. These realms are good for beginners as they will allow you to explore the world uninterrupted by the opposing faction unless you actually step on their territory and go in the range some guards. On PvE realms you can only fight other players if you have PvP enabled. To do so type "/pvp" into the message dialog to enable PvP mode, type the same command again to turn it off. PvP will be disabled after you have showed no PvP activity for 5 minutes.
Role-play realms are for those who want to step into their character's shoes completely, guide their every action according to a certain story and behave like you are that character while talking to other people in the game (except in the general channels). RP realms are not for everyone, so only choose such a realm if you really think you can handle it. RP realms also have stricter policies for naming characters.
Those are similar to role-play realms but are aimed at PvP battles as well.
Read the information about the different <a href="http://www.worldofwarcraft.com/info/basics/realmtypes.html">Realm Types</a> on the official site for further information, rules and policies for each type of realm.