Rondo, and its French equivalent rondeau, is a word that has been used in music in a number of ways, most often in reference to a musical form, but also in reference a character-type that is distinct from the form.
In rondo form, a principal theme (sometimes called the "refrain") alternates with one or more contrasting themes, variously called "digressions, couplets, episodes," or "subordinate themes." The overall form can be represented as ABACADA... The number of themes can vary from piece to piece, and the recurring element is sometimes embellished or shortened in order to provide for variation.
The form began to be commonly used from the classical music era, though it can be found in earlier works. In the classical and romantic periods it was often used for the last movement of a sonata, symphony, concerto or piece of chamber music.
A common expansion of rondo form is to combine it with sonata form, creating a sonata-rondo. Here, the second theme acts in a similar way to the second theme group in sonata form by appearing first in a key other than the tonic and later being repeated in the tonic key. Unlike sonata form, there is no thematic development except possibly in the coda.
Rondo as a character-type (as distinct from the form) equates to music that is fast and vivacious--normally allegro. Music that has been designated as "rondo" normally subscribes to both the form and character. On the other hand, there are many examples of slow and reflective works that are rondo in form but not in character. Composers normally do not identify such works as "rondo."