Scene: One team has kicked the ball to the other team.
The player back to receive the ball starts waving his hand in the air over his head.
That is the signal for a "fair catch." Once a player waves his hand over his head, he is saying that he just wants to catch the ball and will not run with it when he does. The kicking team is not allowed to touch him (or it will be a penalty).
If he catches it: His team gets the ball right where he caught it and no players from the other team are allowed to try to tackle him.
If he drops it (and he has touched the ball): either team can recover the ball for their team.
If he doesn't touch the ball and misses it: Then it's like he never signaled for the fair catch. If his team recovers it, they can try to run with it (and the other team can try to tackle him). If the kicking team recovers it, then the next play (for the receiving team) will start wherever the kicking team touched the ball.