Spectator is the person who is not in the fight, but can watch it from a free view mode, or watching it from a players point of view. Spectator is available in all game modes unless otherwise turned off. In the game, it is technically classified as a team (like red, blue, and spectator). Spectator mode is a great way to record demos of other fighters.
As a spectator, you can cycle which fighters you watch from their point of view by clicking whatever button is bound to +attack (usually mouse1). To detach from that fighter, press your +jump button (usually the spacebar). When detached, you can wander freely about the map. Your secondary fire button will allow you to move at a much greater speed.