Contra dancing is a form of traditonal American set dance, with its roots in the French and English country dancing of the 17th and 18th centuries. The dances are done in longways sets of couples, in which a dancer and his/her partner do a short series of dance figures or moves with each other and another couple, then each couple moves along the set to a new couple and repeats the figures, and so on. The figures themselves, which are simple and easy to learn, are similar to those used in traditional square dances (for example, do-si-do, allemande and swing your partner); note, however, that despite the similar-sounding names, contra dancing has absolutely nothing whatsoever to do with country line dancing.
Contra dances are almost always done to live music, usually Celtic or traditional American reels and jigs. There is also always a caller, who first teaches the figures to be done in a particular dance, and then "calls" the dance as the music is played, reminding the dancers of each figure to be danced. An individual contra dance lasts for 8 - 9 minutes; the evening program consists of a series of these dances, sometimes with an occasional couples' dance (such as a waltz or polka) thrown into the mix. Dancers typically change partners for each dance.
No costume or particular style of clothing is worn, though as the dancing is energetic, lightweight attire and low-heeled shoes are more comfortable.
A video, shot at Contra Carnivale in San Luis Obispo
A Contra Dance Primer A very nice basic description of contra dancing from the Hamilton (Ontario, Canada) contra dance community.
Contraculture From the same group; a more detailed description of contra dancing -- including a simple color diagram of the basic formation.
What is Contra Dance? Gary Shapiro's variation on the above
Country Dance and Song Society (CDSS): Celebrating a Living Tradition of English and Anglo-American Folk Dance and Music since 1915. MCDC is a CDSS affiliate.