In context of the game shotgun , the bitch seat is the seat that is last to be claimed after shotgun is called. It is the worst seat in the car, usually the middle back seat, although any backseat in the car can be the bitch seat if it has, for example, a malfunctioning window or crappy apholstery.
Person 1: Shotgun!
Person 2: Dammit. Shotgun cobain! (back left seat)
Person 3: Shotgun back right hand!
Person 4: Ahh c'mon guys, i always get the bitch seat!
Person 3: Sucked in dude.