Catchers typically throw the ball to third base after a strikeout. Throwing the ball about serves to keep the fielders on their toes by shaking things up. Throwing the ball to third helps the infielders' arms stay relaxed for throwing. Also, runners are usually not allowed to advance beyond second base on a strikeout, so the only choice they have is to return home.
This practice dates back to when there were only three bases per game. In that era, if you didn't want to walk anybody, you'd better be able to get them out quickly because there was no such thing as a hit-by-pitch. Catchers would often throw the ball into left field in order to get the pitcher out in a hurry. That's why we say "thrown for effect" or "for show." Today, with four bases covered per game, it's less important for the catcher to get the pitcher out and more important for him to communicate with the batter before he walks away. However, throwing the ball into left field is still done as a form of entertainment or distraction.
When there are no runners on base, this term is often used to describe tossing the ball around the infield after a strikeout. Following a strikeout, the catcher will often toss the ball to the third baseman, who will then throw it to the second baseman, who will then throw it to the shortstop, who will then throw it back to the third baseman. If the batter strikes out looking, the umpire may call "time" and signal for the pitcher to stop throwing.
This is done as a form of team building. It gives the other players on the field a chance to get into position while at the same time showing that you are not afraid to strike out. This also creates some excitement on the bench since everyone wants to know where the ball is. In fact, following a strikeout, many coaches will use this as a signal for their players to come off the bench to take advantage of any balls put into play after the bat (often called "runners in scoring position").
The ball is said to go into left field after a strikeout to signify that you have no choice but to give up your bat speed and rely on your teammates to help you hit future pitches. This is why following a strikeout, most batters will look toward left field before hitting up against the wall in an effort to find a bit of movement on the ball. If they don't, they might as well be swinging a stick because there's nothing useful they can do with it.
If the batter strikes out, he is automatically out unless the catcher neatly holds onto the ball or the ball touches the dirt. If the catcher fails to catch the third strike, the hitter may attempt to steal first base if it is open or there are two outs. However, if an infielder fields the ball and throws home before the batter reaches first base, this will end the inning and prevent the batter from being awarded any bases.
The rule was created to protect batters from having to face excessive pitches from dangerous pitchers. In those days, pitchers threw very hard and frequently used trick pitches - such as the spitball or corked bat - that made them difficult for most batters to handle. Because of this, the league decided that a batter should be considered out if he reaches on an error or when the catcher misses the ball with his glove hand. This would give him time to recover and get back into the batting order without having to face the same pitcher again.
At this point, the catcher is allowed one final chance to catch the ball, but if he cannot do so, then the batter is awarded a base on balls. While this might not seem like a big deal today because runners are usually safe when they reach first base on a walk or hit by pitch, this wasn't always the case in baseball history.