Spectate (often shortened to "spec") is a term used to describe the act of watching another player playing a beatmap in real time (with approx. 500ms delay). To spectate, open the extended chat and click on a grey user panel.
In order a user to spectate another player, said user will need to have downloaded the exact same beatmap that the player being spectated is playing. If the player being spectated starts playing a new beatmap, and the spectator does not have said beatmap, the new beatmap will be announced in a private #spectator channel. However, if the spectator has an osu!supporter tag, the beatmap will be automatically downloaded via osu!direct and processed for them in-game.