The typical Foster-style slugs sometimes have striations impressed into their skirts, angled to make it look as if they are to impart some spin. They do not, but I think the term "rifled slug" comes from this practice.
All of the slugs have the same function; that is to permit a single projectile to be fired from the shotgun, thus turning it into a sort of substitute rifle. As I stated earlier, rifled shotgun barrels ususally work best with a sabot-type slug.
This begs the question of why would anyone fool with slugs instead of just selecting an appropriate rifle. Some states do not permit rifles to be used for deer hunting. Typically, such states have fairly dense populations and the lesser maximum range of a slug is thought to be safer. So, one reason to use slugs might be because the law requires it. A bird hunter, hunting in an area with problem bears, might want to have a few slugs readily accessible. In defensive or police use, the shotgun is usually loaded with buckshot. A shotgun thus loaded provides a great deal of power to its user, within its effective range, and allows, at optimal patterning ranges, targets to be addressed very rapidly, due to the spreading pattern permitting a bit less precision in aiming. The defensive shotgunner should also have some slugs readily available so that he can engage target which pose a threat and which are beyond the effective range of buckshot (20-25 yards...sometimes less). In a carefully zeroed shotgun, the slug can also be selected for engaging a threat with has innocent or uninvolved parties in close proximity (such as a "hostage" situation).
Slugs can also be used to breach doors by blowing off the hinges and/or lock mechanisms at near contact distances. This is an improvisation, however, and is best conducted with specialty, frangible, "breaching" rounds to minimize danger to persons inside.