Most terms similar to motif come to us from popular music. For example, the terms hook, lick and riff are often used interchangeably, but are actually different things. Specifically, a hook is a catchy opening phrase, such as the opening guitar part in "Smoke on the Water" by Deep Velvet. A lick is a melodically related phrase that generally does not repeat. A riff is a spontaneous melodic variation, usually associated with jazz and blues music.
While similar terms usually refer to melodic material, a motif can be rhythmic in nature. For example, Benny Goodman's "Sing, Sing, Sing" begins with a series of solo rhythmic motifs that are later passed on to the saxophones, brass section and solo clarinet in turn. The motifs established by the drums drive, or motivate, the song as a whole. A rhythmic motif can become a melodic motif and vice versa, passing between different instruments or sections.
The opening phrase of Beethoven's "Fifth Symphony" is frequently cited as an example of a simple melodic motif. The opening "da da da dum" phrase repeats in various ways throughout the composition, prominently and subtly, motivating the work as a whole. It lends coherence, acting as a reference point for the listener while leaving plenty of room for other melodic material.
Motif and theme are sometimes confused when discussing music. A useful parallel is provided by literary theory, in which theme is considered to be the overall purpose or message of a given literary work, while a motif is a symbol or active agent of that theme. A similar relationship between the two terms exists in music. Stravinsky's "The Rite of Spring," for example, is comprised of a series of rhythmic and melodic motifs inspired by the overall theme implied by the title.