A fugure ('flight') is a contrapuntal piece (see counterpoint), in which two or more parts are built or layered on a recurring subject that is introduced alone and followed by an answer, which is the subject (or theme) at a different pitch. The original subject and answer are extensively manipulated, in combination with each other or fragments thereof, and sometimes with a recurring countermelody (the countersubject), traversing several contrasting keys before returning to the tonic.
A fugato is a passage in fugal style or emulating fugal texture, but not instantiating an entire fugue.
The fugue associated most strongly with the Baroque period and particularly with Bach, in which three or more voices enter one after another, not only imitating the main subject (as in a canon) but developing it in new directions and sometimes introducing new material. With the rise of neoclassical music in the 1920s and 30s, fugal writing came back in fashion.