The Official Rules of Major League Baseball section 2.00 states:
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.
Once a batter has recieved one, two, three strikes, he is out. If, however, the catcher is unable to trap the ball following the third strike, and first base is open, the runner may make a mad dash for first base, hoping to get there before the catcher can find the ball and throw to first. Even if the batter does reach safely, the pitcher is still credited with a strikeout.
A strike is signalled by the umpire standing behind the catcher at home plate, called the umpire in chief, or more commonly the plate umpire.
The umpire typically signals that a ball is a strike by pivoting to his left and pointing or jabbing his right arm in a demonstrative fashion. If he is still crouched behind the plate, the umpire will typically begin to stand during the motion. If the umpire is already standing by the time he makes the call, he may instead extend his right leg and kneel slightly as he points. If the strike was particularly forceful or impressive, especially if it resulted in a strikeout, the umpire may choose to precede the right-arm thrust with a left-arm thrust, thus producing a "punch 'em out" effect.
In all cases the gesture is accompanied by some sort of noise. This can be a simple grunt, a clipped, drill-seargant yell of "Huyk", or even an elucidated, obnoxious "Steeeeee-riiiike", which may be followed by the strike count if you are Leslie Neilsen.
No motion or sound should accompany a pitch that is not a strike. The umpire should merely stand and relax while the pitcher and batter go through their elaborate pre-pitch rituals of spitting, staring, bat-swinging, glove- and crotch-adjusting, chest-crossing, bat-kissing, etc.
If the umpire wishes to drive the fans into a murderous frenzy he can, after a pitch has been thrown to the home team's batter, choose to stand and look around idly for a bit, causing the fans to think the pitch was a ball, and then call the pitch a strike in the manner described above.
Upon the request of the batter, pitcher, or catcher the umpire may signal the count of balls and strikes on the fingers of his left and right hands. If the fans, displeased by the strike calls and the manner in which they are made, have begun cursing and throwing beer bottles at the umpire, it is still not acceptable for him to signal a count of "one and one" with his middle fingers.