The NFL released a video explaining what a catch is. It seems to be a debate every single game and more and more people seem more confused about what a catch is and what it is not.
In the field of play, three things have to happen:
1. The player must secure control of the ball in his hands or arms prior to the ball touching the ground AND
2. The player must touch the ground inbounds with both feet or with any part of his body other than his hands AND
3. The player must maintain control of the ball after he has completed steps 1 and 2, until he has clearly become a runner.
If a player goes to the ground in the field of play or in the end zone before establishing himself as a runner, he must maintain control of the football after he hits the ground before the ball touches the ground.
If the catch is at the sideline, going out of bounds, then steps 1 and 2 must be completed and then the player must have "complete and continuous control of the ball until after his initial contact with the ground, or the pass is incomplete."
Here is a rule I didn't know -- if a player catches the ball and, before coming to the ground, is held up by one or more other opposing players and carried out of bounds, it is a catch (or interception). I think back to 2009 when Matt Leinart started for the Cardinals against the Titans and Vince Young threw a fourth down touchdown pass to win the game. I always thought the Arizona DBs should have grabbed the WR who caught him and kept him from touching the ground, carrying him out of bounds. This explains why I have never seen any one attempt this -- it would be considered a catch.
In the case of a "simultaneous catch", where both the offense and the defense catch and retain possession of the ball, the passing team retains possession.
So...the big question. Applying these rules to the Dez Bryant non-catch in the playoffs two years ago, would it have been a catch? Would he have been ruled as having established himself as a runner, or would he have been ruled as falling down? That is where the difference is. If he established himself as a runner, it was a catch. If he didn't, then it was not.