A post-possession foul will result in a penalty for a team on the return side of a possession changing kick. In order for the penalty to be called, the ball must have crossed the line of scrimmage, and the receiving team must retain possession of the ball after it has been kicked.
When the ball changes possession via a kickoff, punt, or missed field goal, there are many fouls and infractions that may occur. The officials on the field must take special notice of if and when the ball changes possession in order to call many of these fouls accurately. For example, if a player on the punt return team gets beat down the field by a player on the kicking team, and then blocks said player in the back to clear a path for the return man, the penalty would be considered a post-possession foul. This is because the offending player's team had already gained possession of the ball, and the ball had crossed the line of scrimmage. If his return man were to fumble the ball during the return, and his team lost possession, a penalty would still be assessed, but it would not be considered a post-possession foul.