Understanding the relationship between ticks and beats is helpful if you plan on getting your MIDI recordings to show up in the SCORE editor in a meaningful way. As Mark stated, based upon a measure in common, 4/4 time, each of the four beats (the four quarter notes) has 960 "ticks" -- the entire bar (all four beats) has 3,860 ticks.
When you record a MIDI track (either by playing it in on a MIDI keyboard or placing it onto the Piano Roll editor), you can see how close to each "downbeat" you have played by opening the event list. You can edit "length" (i.e., the number of ticks) in order to make what you intended to play display correctly in the score editor (however, this will alter your performance).
There appears to be some confusion between what you are thinking of in terms of a "clock tick" and what is otherwise considered "one beat" (i.e., a quarter note). To recapitulate Mark, it follows that if one quarter note has 960 ticks, then one sixteenth note has only one-fourth as many...or 240 ticks. Maybe thinking of ticks as pulses-per-quarter note, or "PPQ", would be easier.
In complex time signatures, or in very fast or slow tempos, the relationship between "beats" and "ticks" may be blurred. The tempo you set in Logic should (AFAIK) determine the number of ticks of resolution.
