A situation in which the batter has been charged with a third strike, but is not (yet) out because the catcher failed to catch the strike-three pitch on the fly. A dropped third strike situation can occur any time first base is open (not occupied by a runner), or there are two outs. When this occurs, the batter may (in fact, is forced to) try for first base, and the defense must put him out via a fielding play, as if the batter had hit the ball.
If first base is occupied by a runner, and there are less than two out, the batter is out on the strike-three call and may not try for first base. This is to prevent the defense from getting an easy double play by means of the catcher intentionally dropping the third-strike pitch in front of him and throwing to second base to force out the runner, after which an infielder throws to first to put out the batter. Of course, with two out, a double play is not possible. In this situation, the defense may opt to put out a forced runner rather than the batter.
Often, if the catcher fails to catch the third-strike pitch but keeps it in front of him, the batter will not bother running, since the odds of reaching base in this situation are very near to zero. This is why you will sometimes see a catcher tag a struck-out batter as he leaves the batter's box -- the catcher dropped the ball, and has to tag the batter to complete the putout.
Scoring: if the batter reaches first base safely, and no other runner is put out, then assuming no fielding error occurred, the official scorer must score the play as either a wild pitch or a passed ball, as the case may be. If in the scorer's opinion, the batter or a runner would have been out except for a fielding misplay, then the scorer charges the play as an error, in the usual fashion. In all cases, the batter is charged with an official at-bat. In all cases when the batter or a runner is put out, the play is scored as it actually occurs, instead of being automatically credited to the catcher, as is the case with ordinary strikeouts. The pitcher, however is still credited with a strikeout, even if the batter reached first base safely. This can result in a situation where a pitcher is credited with more than three strikeouts in an inning. A pitcher being credited with four strikeouts in an inning happens, on average, 3-4 times per year in the Major Leagues. Five strikeouts in one inning has never happened in a Major League game, but it has happened a few times in the minor leagues.