The verb to analyse means to examine the elements, components or structure of something in detail. In chemistry, for instance, if you analyse a substance it means that you break it down into/ identify all its component parts. If you analyse a text, such as a poem, you work out how the words are put together to achieve certain effects. The noun form is analysis; you can either "analyse" something or "do an analysis" on it; the meaning is about the same.
The word comes from the Greek analuein, "to unloose or untie," which in a way is similar to its modern meaning.