Staccato
Staccato is defined as the mode of play in which the finger is quickly bounced off the key so as to produce a brief sound with no sustain. It is somewhat astonishing that most books on learning piano discuss staccato, but never define what it is! This paragraph gives a fairly complete definition. In staccato play, the backcheck is not engaged and the damper cuts off the sound immediately after the note is played. There are two notations for staccato, the normal (dot) and hard (triangle). In normal staccato, the jack is generally released; in hard staccato, the finger moves down and up more rapidly and generally leaves the key before the jack is released. Thus in normal staccato, the key drop may be about half way down, but in hard staccato, it is generally less than half way. In this way, the damper is returned to the strings faster, resulting in a shorter note duration. Because the backcheck is not engaged, the hammer can "bounce around", making repetitions tricky at certain speeds. Thus if you have trouble with rapidly repeated staccatos, don't immediately blame yourself -- it may be the wrong frequency at which the hammer bounces the wrong way. By changing the speed, amount of key drop, etc., you may be able to eliminate the problem. Clearly, in order to play staccato well, it helps to understand how the piano works.
Staccato is generally divided into three types depending on how it is played: (i) finger staccato, (ii) wrist staccato, and (iii) arm staccato. (i) is played mostly with the finger in a pull motion, holding the hand and arm still, (ii) is neutral (neither pull nor thrust), played mostly with wrist action, and (iii) is usually best played as a thrust, with the playing action originating at the upper arm. As you progress from (i) to (iii) you add more mass behind the fingers; therefore, (i) gives the lightest staccato and is useful for single, soft notes, and (iii) gives the heaviest feeling and is useful for loud passages and chords with many notes. (ii) is in between. In practice, most of us probably combine all three. Some teachers frown on the use of wrist staccato, preferring mostly arm staccato; however, it is probably better to have a choice of all three. For example, you might be able to reduce fatigue by changing from one to the other.
Because you cannot use the arm weight for staccato, the best reference is your steady body. Thus the body plays a major role in staccato play. Speed of staccato repetition is controlled by the amount of up-down motion: the smaller the motion, the faster the repetition rate.