1. to separate (a material or abstract entity) into constituent parts or elements; determine the elements or essential features of (opposed to synthesize). 2. to examine critically, so as to bring out the essential elements or give the essence of: to analyze a poem.
Analyze is the American spelling of the same word. It is a verb, and can be used in all the same contexts as analyse. You can see in the following graphs that analyse is much more common in British English, whereas analyze is much more widely used in American English.