Derek Shepherd is a renowned brain surgeon who we are led to believe moved to Seattle from New York a little bit before the series debuted. He and his intern, Meredith Grey, met and slept together during the Pilot, before they even knew each other's identity, with their romance blossoming soon after.
Derek is married to Dr. Addison Montgomery Shepherd, who he left in New York after finding her in bed with his best friend Mark Sloan, a plastic surgeon. Addison joining the staff at Seattle Grace during Season Two resulted in constant conflict for Derek as he struggles to repair his marriage and deal with his feelings for Ms. Grey. Ultimately, he goes for his true love.
Dr. Shepherd was nicknamed "McDreamy" by the female interns at Seattle Grace. The man has some amazing hair. He and Dr. Preston Burke (while he was around) were among the hospital's top surgeons and there has been occasional friction other which one the chief, Dr. Richard Webber, would name as his successor.
Derek was named Chief of Surgery by Richard at the end of Season Three, but he did not accept. He and Addison got a divorce after he rekindled his love for Mer, but he has expressed doubts over that relationship's chances of lasting, as well. As he put it, he doesn't know if he can keep breathing for her.
We'll see what Season Four holds for these two.