A boron atom acting as an acceptor in a silicon lattice. Source
An acceptor is kind of a complicated idea. The simples explanation is that it is an impurity purposefully put into a semiconducting material because it has fewer valence electrons (outer shell electrons, the ones that bond to other atoms) and will thus create an electron hole that will move around the lattice as electrons jump to repair the hole, only to make the hole again where they jumped from. This hole can carry a charge, and thus is useful.