Hapkido is a Korean martial art which is not widely known in the U.S. It's popularity is eclipsed by its cousins, Tae Kwon Do and the Japanese art of Aikido.
Hapkido was founded earlier in this century by Choi, Yong Sul. Grandmaster Choi had trained in Daito Ryu Aikijujitu for many years in Japan before returning to his native Korea. There he began teaching his art which he initially called Hap Ki Yoo Sul, Korean for "Akijujitsu". Some of his students were also knowledgeable in traditional Korean styles, with their heavy emphasis on kicking, and these were combined creating a new style, Hapkido.
Modern Hapkido is very diverse with many different substyles and offshoots. There is no overall governing body though there are a couple of good sized organizations (American Hapkido Association or the World Hapkido Federation to name two) which have attempted to fill this role. Unfortunately the different groups are often at odds and so the lack of cohesion within the art continues. In fact, the brief history of the art given above is not always agreed upon by the various groups. Nevertheless, there is a basic similarity between most of the branches of Hapkido and most students will feel comfortable in any of the organizations.
Hapkido is a generalist style. Many martial arts are specialist styles, for instance Tae Kwon Do which focuses on kicking and striking, with heavy emphasis on the kicking or Judo which focuses on throwing and grappling. Hapkido combines kicking, striking, throwing, joint locking, grappling, and pressure point techniques to create a broad and effective style of self defense.
My personal training in Hapkido is from the University of California, Berkeley Martial Arts Program. Since leaving Berkeley I have continued to train and I currently teach Hapkido at Brown University and also Harvard University.