When is the ball or player considered down in NFL football?

The runner is considered down when either a: His forward progress stops or b: any part of his body other than his foot or hand touches the ground AND he is touched by a player of an opposing team.

There is no "down by contact" in College Football. That's only in the pros. When a college player goes down he cannot get up and continue running, period. He does not have to be touched by an opposing player to be considered down.