Reading the other responses here it occurs to me how remarkable it is that so much needs to be defined to fully understand this one pitch. Minimally, you need to know what an "out," a "run," an "at bat," a "ball," a "strike," a "base," a "base hit," and a "walk" are.

First of all, let's understand what the objective is. The offense, that is to say, the batter, is attempting to advance around each of the four "bases" in counter-clockwise order (i.e. score a "run") without being eliminated from play (i.e. make an "out"). The offense is trying to score as many runs as possible before making three outs. Teams alternate turns attempting this.

The batter most typically advances by striking a pitched ball within play such that they are able to safely reach base without making an out. This is called a "base hit." An out is made on balls in play when the ball is caught cleanly off the bat in the air, when the runner is tagged by the ball while not on a base, or when the ball reaches a base ahead of the runner in a situation where that runner is not allowed to retreat to the previous base. This happens when another runner from the same team occupies the base behind the runner (and always on any play between home plate and first base). This last case is called a "force out." (Existing runners also must not attempt to advance on balls hit in the air until they have been caught by a fielder. This rule prevents a runner from setting off on a sprint while a high fly ball lazily floats through the air and into a fielder's glove.) [0]

A given play ends when all runners are accounted for -- that is, when each runner is either safely on a base or has been eliminated.

A batter has limited opportunity to put the ball in play each time he goes to bat (an "at bat"). A zone is defined -- the "strike zone" -- through which any pitched ball not swung at results in a penalty for the batter -- a "strike". This area stretches from left to right over home plate and vertically from roughly the batter's knees to the letters on his chest. However, any swing that misses the ball also results in a strike, no matter where that ball was pitched. Finally, a batted ball that lands out of play -- a "foul ball" -- can also result in a strike. However, such a batted ball can never result in an out (unless, as with balls in play, it is fielded cleanly in the air). It can never be the final strike in an at bat. [1]

Any pitched ball that does not result in a strike is counted as a "ball". A hitter receiving three strikes is called out (i.e. he "strikes out"). A hitter receiving four balls is rewarded first base -- a "walk." In either case, his time at bat is complete.

The batter may also be rewarded first base when a pitched ball strikes his body. However, provided he does not hit the batter, the pitcher is free to throw the ball anywhere he pleases, with the obvious incentive to either make the batter swing and miss or to place the pitch within the strike zone without him swinging (thus (hopefully) generating a "called strike" from the umpire).

So, this pitch is legal and the batter in this case is indeed penalized. He swung and missed, which is always a strike, no matter where the pitch was located. However, it is a minor penalty. This was the first pitch in this at bat and therefore his first strike. He remains at bat with two more strikes remaining to attempt to reach base safely in one of the ways described above.

[0] The runner is actually free to start toward the next base on balls hit in the air, but he must return to his starting base if the ball is caught. He is out if the defense can tag his starting base with the hit ball in hand before he is able to return.

[1] A ball fouled directly back into the catcher's mitt is indeed a third strike (and, thus, an out). It is treated the same as a swing and a miss.