A realm is an instance of the World of Warcraft (WoW) game world. Realms are hosted on physical devices called servers (for this reason, the terms "realm" and "server" are often used interchangeably).
To play WoW, you must first choose a server (realm) to play on. You then connect to this realm over the internet. Blizzard has servers in the US, Europe, China, and Korea (see Localization).
Why have realms?
The WoW game world is duplicated (mirrored) across multiple realms to prevent any one realm from becoming too crowded (and thus overwhelming the server), to provide for better connection times (by having players connect to local servers), to allow for localized communities based on language (for example, French speakers on realms in France), and to appeal to different play-styles for different users.
To allow for different play-styles, realms come in four types:
PvE (a.k.a. Normal) - Player vs. Environment
PvP - Player vs. Player
RP - Role-Playing with PvE
RP-PvP - Role-Playing with PvP
For the adventurous, there is also a Test Realm that is periodically available in which players can try experimental features in development before they get released as general patches.