So what happens at kickoff?

The game starts with the kickoff - when one team kicks the ball to the other. (There are other times when there are kicks throughout the game, too - and the same choices apply.) You'll notice one guy back to catch the ball near the endzone. There's a few choices he has when the ball is kicked...

1. Catch the ball and run it. He can try to run it as far as possible (while the other players try to tackle him to make him stop).
2. Waive his hand above his head in the air to signal a "fair catch". This means he is going to catch the ball, but he can't run with it and the other team can't hit him. However, if he drops the ball, then it is anyone's ball and either team can get possession of the ball by being the ones to grab it and hold on to it.
3. Let the ball bounce and not try to catch it. Then, wherever the ball stops bouncing/rolling, that is where the next play will start. Or, if the kicking team touches the ball, that's where the ball is "dead" and that's where the next play will start.

If the kicker kicks the ball all the way into the endzone and the player doesn't try to catch it and run (or if he catches it and "takes a knee"/kneels while holding the ball, then it is considered a "touchback" and the next play will start at the 20-yard line).

On a kickoff, if the kicker kicks the ball out-of-bounds on the sidelines, then the ball automatically starts at the 40-yard line! So those kickers are pretty careful to kick the ball straight down the field...