He is the father of the protagonists, Edward and Alphonse Elric. He is a brilliant and well-known alchemist. He left Ed, Al, and their mother Trisha when Al was a baby and Ed was very young. Years later, Ed still holds a grudge against him for abandoning his mother, which he believes contributed to her death.

(Source: Wikipedia)

In the Fullmetal Alchemist manga he is refered to as "Van Hohenheim," not Hohenheim Elric. As Edward Elric points out in book 14 of the Fullmetal Alchemist manga, "Elric is my mother's name!"
He (in the original 2003 anime)...

Has been alive for centuries. He worked in a lab, which took advantage of 'witch trials' as human bait, to create the first philosopher's stone. He succeeded, but it cost him his life. Dante, (his lover at the moment) attached his soul (much like Al to armor) to the nearest man in the room, reviving Hohenheim. They continued this practice for many years, until Hohenheim fell in love and married Trisha Elric. After they sired Edward and Alphonse, he left for fear Dante would use them against him, and also because he felt ashamed of his past. Like Dante, his flesh is wearing away, so he must wear cologne to mask the scent. He can transmute without circles, because when he died, he saw past the gate, as did Dante.


He (in the manga and 2009 series) is...
a human Philosopher's Stone. Homunculus (Father) was a test subject at a lab in which Hohenheim worked (in fact, Father was the one who named Hohenheim). The king of Xerxes demanded the secret of immortality from the homunculus, and he obliged. a country wide transmutation is formed, and everything is working as planned. However, at the last minute, Father lies about where the center of the circle is, and engulfs both himself and Hohenheim into the gate. Hohenheim wakes to see everybody in his country dead. He meets Homunculus, who tells him everything, how he has now gained half of the powers of the stone. Homunculus did this out of gratitude. after these events, Hohenheim travels to Xing, and founds the art of Alkahestry.