In the NFL, if a player falls down or trips on his own (and stays in bounds) and is not touched by a player of the other team, he can get up and keep the play going. If a player of the other team causes him to fall (tackle, pushes, etc.), then when the player falls, he is considered "down by contact" and the play is over.
To know if he is officially down, they look for an elbow or knee (or butt) to touch the ground. If just his hand hits the ground, he's not considered down and can keep moving the ball.
Sometimes, the player with the ball will fall on top of a player from the other team. If his knee or elbow or butt doesn't touch the ground, he can keep going!
You will see these kind of plays reviewed or challenged too - that's when they're looking to see if someone on the other team touched him (sometimes it's just a whiff and no contact was made) and if his knee/elbow/butt actually touched the ground.