Slugs are molded chunks of metal, nylon or plastic. In effect, they turn a shotgun into a crude rifle. Slugs are fired individually, like bullets, instead of in bunches like buckshot and birdshot. They can come in a variety of shapes, but they are often tapered into a bullet shape. They can be solid or filled with substances like explosives or incendiary powder.
Shotgun slugs can be rifled -- this is supposed to make them spin in the air and thus improve their flight length and accuracy.
One reason hunters use slugs is to hunt deer in states that ban the use of rifles and/or buckshot ammo. The shotgun/slug combination provides a legal, if shorter range alternative. There are at least 20 states that have restrictions of this kind.
Non-explosive slugs are also used for crowd control. When deployed properly, they can act as a non-lethal deterrent in these situations. They are used in organized shooting competitions as well.