Any ball carrier who is either tackled to the ground by the opposing team, or who falls to the turf on his own and is touched by the opposing team. Certain body parts such as the knee, elbow, hip, and head are considered down by contact. Basically any body part other than hands, fingers, and feet constitue a ball carrier to be declared down.
The down by contact rule does not apply to the college football ranks. Any college ball carrier who falls to the ground on his own is considered down, and the play is over. In the NFL, a ball carrier needs to be touched to be considered down. The one exception: if an NFL quarterback decides to run and slides to the turf feet first, he is considered down where the slide begins.