A premise is what forms the basis of a theory or a plot. When you called 911 on the guy in your back yard, it was on the premise that he was a thief and not the meter-reader.
In logic, the premise is the basic statement upon whose truth an argument is based. Criticize alternative theories by demonstrating their false premises. In a more general sense, it's a basic assertion. On the premise that people are generally good, you keep your doors unlocked. Premise is also used more loosely for the starting point for a plot. A movie's premise could be that two people trapped on a desert island will inevitably fall in love.