When using the term geek in reference to circus performers it is possible to be slightly more specific on the nature of the act. A geek was a performer who consumed unusual or unpleasant substances while on stage. Specifically the geek was often billed as a "wild man" who would bite the heads of chickens, consume broken glass, eat nails, etc. Their appeal of course lay in both providing an example of extreme behavior and in breaking social taboos regarding consumption.
Today this sort of act has, regrettably, fallen out of favor with most mainstream circuses touring the United States. In a continuing effort to provide family entertainment modern circuses, for the most part, employ mostly conventional acts involving trapeze work, trained animals, tumbling, and clowns. However some circuses still employ these sorts of older acts, including the Jim Rose circus. If you view the episode of the X-files that features performers from the Jim Rose circus you can see an example of a circus geek, in this case the gentleman with the blue puzzle pieces tattooed over his entire body.
Information in part provided by the Unix online Webster's dictionary and the Official Jim Rose circus webpage.