The defensive offside penalty exists in football to punish defensive players for attempting to get a head start on the play. If this penalty wasn't strongly enforced, defenders could get started on their pursuit of the ball carrier before the offense can leave their stance. That doesn't seem very fair which is why this penalty exists.
The defensive offside penalty occurs when any part of a player's body is on the other side of the line of scrimmage as the ball is snapped. While this can happen to offensive players, it is mostly called on the defense. Unlike the offense, defensive players don't need to be set in their stances and are more likely to jump early or naturally move offside before the snap of the ball.
Defensive linemen and pass rushers are most often called for being offside because they are on the line of scrimmage and move across it as the play starts. Other defensive players are moving backward or off the line of scrimmage entirely.
In the NFL, the play continues despite the offside penalty. Quarterbacks like Aaron Rodgers are famous for using a pending offside penalty as a chance to gamble and throw deep. If it's an incompletion or worse, teams take the penalty and if they complete the pass, they decline the penalty and take the yards. It's a win-win situation for great quarterbacks.
The NCAA, CFL, and AFL are also like the NFL where they allow some offside penalties to be live ball fouls. The NFL seems to be the league that allows play to continue most especially because the quarterbacks are good enough to take advantage of it.
It's unclear on how often offside penalties are live ball fouls in high school football. This ruling might even differ state to state.
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To signal an offside penalty, the referee will bend their arms and put it at their hips. The referee then usually points in the direction of the defense and says the number of the player who committed the penalty. The offense can get called for being offside too, but that occurs much less often.