A Boss is a powerful, normally unique enemy in Dark Souls, which doesn't respawn once killed. Bosses are distinguishable from normal enemies as their name and HP are displayed at the bottom of the screen. Bosses are usually found on the opposite side of a white Fog Door.
While players will generally end up fighting most of the bosses, not all of them need to be killed in order to beat the game.
As well as bosses there are a number of other enemies which don't respawn upon death (sometimes referred to as minibosses).