Karen Ann Quinlan.
Born March 29, 1954.
After a party on the night of April 15, 1975, she ceased breathing for at least two fifteen-minute periods. Her friends tried artificial respiration, but apparently didn't know how to do it properly, and before she reached the hospital Quinlan suffered heavy brain damage from lack of oxygen.
She wasn't entirely brain dead, however -- she showed some rudimentary movement and reactions. Because her brain was not completely dead, the hospital refused to allow her to be taken off a respirator. Her father sued to be allowed to take her off the respirator so that she could die naturally, since there was no hope for any sort of recovery for her.
He lost the first case, but appealed and won in 1976 in the Supreme Court of New Jersey case.
Further details on her medical and legal case can be found here.
Ironically, after she was removed from the respirator, Quinlan had just enough to brainstem function to be able to continue to breath (poorly) on her own, and she survived another ten years, until she died on June 11, 1985 due to pneumonia.