There are two different ways a player can be down by contact. First, if the player has any body part other than his hands or feet on the ground and a defender makes contact with him by touching him in any way. The second definition is if the ball carrier hit the ground due to the action of a defender. This means that the defender pushed or tackled him hard enough that the offensive player is considered down, even if no one touched him while he was on the ground.
In the NFL a player can stumble or trip and land on the field but the play will still be alive. If there is no initial contact from a defender or a touch while he is on the ground, the ball carrier is able to get up and continue running.
The biggest debates regarding down by contact rulings are related to fumbles. The most common criteria officials are looking for when they review these turnovers is if the ball was knocked out before the runner was ruled down by contact. Many times the ball popping loose and the knee or elbow hitting the ground will be almost simultaneous, and the referees will have to execute their best judgment as to whether the play was officially over before the fumble.
The other down by contact ruling that occurs multiple times throughout an NFL season is whether a player was down before the tip of the ball crossed the goal line on a touchdown attempt. Although sometimes this call is easier to make, when running backs or full backs dive through the linemen on a small yardage push, it is often hard to see through all of the bodies as to whether any body part illegally hit the ground before the ball crossed the plane.