Hapkido is a Korean Martial Art that was created out of the Japanese Martial Art of Daito Ryu Jujutsu.
Hapkido was founded in 1963 by Choi Young-Sul. Choi was born in Korea in 1904 and taken to Japan when he was a child. He became a servant, of Daito Ryu Aikijujitsu Grandmaster Takeda Sokaku. Choi was allowed to train in Daito Ryu Aikijutsu and trained in it for 30 years.
Choi returned to Korea in 1945, after Japan's defeat in World War II ended the Japanese occupation of Korea. In Korea, Choi trained with Black Belts in Korean Martial Arts and added kicking techniques from Tae Kwon Do into this art that he renamed Hapkido.
It could be said that Hapkido is a combination of Japanese Daito Ryu Akijujitsu, (a form of Jujitsu that emphasizes Throwing and Joint Manipulations) and Atemi (Strikes to Vital Areas of the body, combined with Knee Strikes, Elbow Strikes and a study of Pressure Points).
Add Tae Kwon Do's Kicks and Punches and you have a pretty complete Martial Art.
Hapkido's techniques continue to evolve and today also incorporate punches from Boxing, and it also contains some aspects of Grappling. Hapkido is not a sport but a Combat Martial Art designed to kill or disable an enemy or attacker.
Today there is no one style of Hapkido, each instructor of Hapkido blends their favorite techniques into their own version of the art.