Q:

Ball goes through the strike zone, but the catcher doesn't catch the ball?

I'm still learning the basics of baseball, so please excuse my ignorance. 
If a ball is pitched through the strike zone, but the catcher doesn't catch the ball, does this count as a ball or strike?

A:

According to MLB's definition of terms, a strike is a pitch in which "any part of the ball passes through any part of the strike zone" and called by the umpire, regardless of whether the catcher catches the ball or not. 
An instance in which this is significant is when it is the third strike. If the catcher doesn't catch the third strike, the batter can become a runner (see rule 6.09 (b) quoted below). Base runners, if any on base, can also attempt to proceed when the catcher doesn't catch the ball, but this is more common on wild pitches.

A STRIKE is a legal pitch when so called by the umpire, which -- 
(a) Is struck at by the batter and is missed;  
(b) Is not struck at, if any part of the ball passes through any part of the strike zone;  
(c) Is fouled by the batter when he has less than two strikes;  
(d) Is bunted foul;  
(e) Touches the batter as he strikes at it;  
(f) Touches the batter in flight in the strike zone; or  
(g) Becomes a foul tip.

6.09 
The batter becomes a runner when -- 
(b) The third strike called by the umpire is not caught

providing (1) first base is unoccupied,
or (2) first base is occupied with two out;

