What is a fair catch? A fair catch is a signal that the player catching the ball from a kickoff or punt uses to tell the other team and referees that the offense is choosing not to run the ball. He waves his arm over his head like in the image below. Once he's signaled for a fair catch, no defender is allowed to tackle him and the offense will start playing where the ball landed. If a defender accidentally tackles him after he's signaled for a fair catch, it is a big 15-yard penalty, meaning the offense gets to start 15-yards closer to the goal line.
Why signal for a fair catch? The kicker or punter is trying to kick the ball as far as possible down the field. He wants the ball to hang in the air as long as possible so his other teammates can run down the field and start defending against a big kick/punt return. While the kick/punt returner is waiting for the ball to drop, he gets a chance to scope out where the defenders are. If they are all closing in and there doesn't seem like a great opportunity for a run, he'll signal for a fair catch. That way he avoids getting tackled. Some of the biggest injuries can happen on kick returns because the defenders have been running at full speed from many yards away. They build up momentum and a big collision/tackle can have dangerous results.