American football term for when a player loses possession of the ball and it hits the ground. From this point, the first player to gain possession ("recover the fumble") gets control of the ball for his team.

In order for a receiver to fumble the ball, he must be in complete control of the ball first. In other words, if a quarterback passes to a receiver, and the receiver is juggling the ball and does not have control, then gets hit and loses the ball, it would not be a fumble (it would instead be an incomplete pass).
However, on non-passing plays, the offensive player does not need to have control of the ball for there to be a fumble (for example, a botched snap from the center to the quarterback would be a fumble).

If the ball is fumbled out of bounds, the team who fumbled it retains offensive possession. If the offensive team fumbles the ball out of their own endzone, the defensive team would earn a safety (two points). If the offensive team fumbles the ball out of the defensive team's endzone or if the defensive team recovers the ball in their own endzone, the result would be a touchback (the defensive team getting the ball on their own 20 yard line).

As with most sports rules, it's harder to explain than it is to understand when actually watching or playing...