Diablo III is not like WoW, it is more a single player than a multiplayer game. If you play in single player, then there are no other players in "your" world (however, an internet connection is required). You might group with friends to join a world together or you can join a public game. In both cases, there are several other players in your world, but not like WoW, because there are only few players in the same realm.
You don't have to choose a realm for a character, though a character is only available in one region (America, Europe, ...).
Hope I could help you.