She has the ability to unlock and/or open any and all kinds of doors, and virtually anything else that can be opened or unlocked. On at least one occasion, injured and cornered, she defended herself by “opening” a man’s chest and removing his heart. This seemed to be done only out of desperation, however. She can also create doors where there are none, and use them to teleport.
The rest of her family was murdered by two assassins, Croup and Vandemar, in “Neil Gaiman’s Neverwhere“. They were hired by the angel Islington, who’d been exiled from heaven and wanted to force Door into creating a portal back to paradise.