For other meanings of the word "canon" see canon.

A canon is a piece of voices (or instrumental parts) that sing or play the same music starting at different times. A round is a type of canon, but in a round each voice, when it finishes, can start at the beginning again so that the piece can go “round and round”. Perhaps the most well-known canon is the Pachelbel Canon written by Johann Pachelbel.

Different types of canon

There are different kinds of canon.  Canons can be described according to distances between the entries of the voices.  If the second voice starts one bar (one measure) after the first voice, this is called a “canon at the bar”.  If it starts after only half a bar, it is called a “canon at the half-bar”.  It is even possible to have very close canons, e.g. “canon at the quaver (eighth note)”.  Olivier Messiaen wrote a 3 part canon at the quaver in his ''' for violin and piano.  The pianist’s right hand (playing chords), his left hand and the violinist are the three parts.

Canons can also be described according to the intervals between two voices.  If one voice starts on a C and the next voice starts the same tune on an F above this is a “canon at the fourth” (because the interval (distance) from C to F is called a “perfect fourth”).  If the second voice has the tune upside down (inversion) this is called “canon in inversion”.  If the second voice has the tune at half the speed (each note being twice as long) this is a “canon in augmentation” or an “augmented canon”.  The opposite is a “canon in diminution”.

“Strict canon” means a canon where each voice imitates the first voice exactly all the way through the piece.  If this does not happen (i.e. if it starts off as a canon but then becomes freer) it is “free canon”.   A canon may start off sounding like a fugue, but fugues have their own form and rules.

Canons in the history of music

Canons were already popular in the 14th century when composers enjoyed writing music for several voices in which each voice has a share of melody (this is called polyphonic music.)   Composers like Guillaume de Machaut wrote canonic music.  His Sanz cuer m’en vois is a three-part canon in which each part has different words.

Probably the greatest writer of canons in the eighteenth century was Johann Sebastian Bach.  Many of his organ works have canons.  He wrote a famous set of Canonic Variations on "Vom Himmel hoch da komm' ich her" which have several canons at different intervals and inversions.  He probably  wrote this to show young composers how to write good canons.  Bach also wrote a work called the Musical Offering'' which has what he calls a “canon per augmentationem contrario motu” (canon in augmentation and contrary motion i.e. backwards) as well as a “canon per tonos”.  This last one is a modulating canon which means that the tune changes key.  This is hard to compose well so that it sounds good because when the first voice has just changed key the other one is still catching up in the other key.

Bach was a great master at writing canons and other very complicated musical forms.  After 1750 composers became less interested in writing music which was all polyphonic, although many composers still showed an interest in counterpoint.  Haydn, Mozart and Beethoven all wrote canons and even Romantic composers like Schumann and  Brahms showed an interest.  César Franck wrote a canon for the fourth movement of his Sonata for Violin and Piano.  It is quite easy to hear this canon, because the violin plays the tune an octave higher than the piano, and the piano holds on to a long note every other bar while the violin catches up.

In the 20th century composers such as  Schoenberg who wrote serial music were fond of canons.  Modern composers like Pierre Boulez have written rhythmic canons: canons in which, for example, the rhythm of one part is the retrograde (backwards) version of another.

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References
The New Grove Dictionary of Music and Musicians ed. Stanley Sadie: 

Musical forms