A clan is a group of people who agree to play together.
Sometimes it's as simple as a few friends making a sort of video-game gang, sometimes it goes as far as being similar to a sports team, competeing in clan battles, where one clans fights another.
They're very common in every multiplayer video game, and most modern video games support them with clan icons, prefixes, and actualy clan battle modes. BRINK does not have this, however.