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Harmonic rhythm is the manner, speed and variation in the way harmonies change during a piece of music. Most music uses harmonic rhythm to some extent or another, at some point or another. Bear in mind that harmony may be implied as well as directly present - a Bach Sonata for Solo flute contains as much 'harmony' as a Brandenburg Concerto, and therefore also will contain harmonic rhythm.
It may worth pointing out that, in general, western classical music makes more use, and more sophisticated use of harmonic rhythm than other musics. Learning to understand and manipulate harmonic rhythm is not in itself necessary, but it is certainly a useful tool.
First, if you are browsing from ARU, you might want to look at the Grove article: Grove article on Harmonic rhythm.
A good example of the straightforward use of harmonic rhythm is the Bach Chorale. These hymns generally use very stable harmonic rhythms - usually one distinct harmony per beat.
Main text (not usually available until after the lecture).