The defensive team can appeal certain plays to alert the umpires of infractions that would otherwise be allowed without the appeal. Appeal plays are not the same as a manager asking the umpire for an instant-replay review.
An appeal can be made when the offensive team bats out of turn. Baserunning instances that are subject to appeal include a runner failing to tag up correctly on a caught fly ball, a runner failing to touch the bases in order -- either when advancing or retreating -- a runner failing to return to first base promptly after overrunning or oversliding it, and a runner failing to touch home plate and making no attempt to return to it.
No runner may return to touch a missed base after a following runner has scored. And when the ball is dead, no runner may return to touch a missed base after he has touched a base beyond the missed base. That means a batter who hits a ground-rule double or a homer can be called out on appeal if he misses first base and doesn't correct his error before touching second.
Appeals must be made before the next pitch or attempted play, or before the entire defensive team has left fair territory if the play in question resulted in the end of a half-inning. The appealing team must make clear their intention to appeal, either via verbal request or another act that unmistakably indicates its attempt to appeal.