To analyse (analyze is U.S. English) something is to consider it in detail so you can figure out its workings or meaning.
Analyse comes from Greek roots meaning "loosen." If you analyse something, it's as if you're untying it and letting the different parts separate so that you can study them. If you've got some mystery substance, you can analyse it by performing chemical reactions to break it into its parts. If you analyse a poem, you look at it word-by-word and even sound-by-sound. But remember that analyse is a British spelling, with analyze as the American version.