A former NFL rule that was introduced in 1999, it states that when an offensive player is holding the ball to make a forward pass, an intentional forward movement of his arm is considered the start of a forward pass. Therefore even if the player loses possession of the ball while changing his mind and bringing the ball back to his body - it's ruled an incomplete pass.
On March 20th, 2013 the NFL Competition committee voted to eliminate the Tuck Rule, as a result if an offensive player loses possession of the football while attempting to bring it back to his body, it will be ruled a fumble.
Coined the "Tuck Rule Game", on January 19th, 2002 the New England Patriots famously benefited from the Tuck rule during a 2001 AFC divisional playoff win over the Oakland Raiders. Quarterback Tom Brady was sacked and the ball was simultaneously dislodged by Raiders defensive back Charles Woodson, but the referee determined that Brady's arm was bringing the ball back to his body and ruled the play an incomplete forward pass. The resulting call allowed the Patriots to tie the game, force overtime, and eventually win (and go on to win the Super Bowl).
The tuck rule struck again in a 2013 playoff game between the Denver Broncos and Baltimore Ravens. Although this time when Broncos quarterback Peyton Manning was hit in the arm after beginning to throw the ball, it was ruled a fumble (not an incomplete pass as per the tuck rule) and they went on to tie the game 5 plays later. They eventually won the game in double overtime (and won the Super Bowl).