Analyze means to study or examine something carefully in a methodical way. If you analyze your child's report card, you may determine his strength and weaknesses (and how many times he cut class).
This verb analyze comes from the noun analysis. The noun analysis was in turn borrowed from Greek, from analyein, or "to dissolve." If you go into analysis, it means that you will be examined and helped by a mental health professional — who will analyze your particular problems and help you find solutions.