If a car bumps another car in the back from behind then what is the part that gets damaged called?

How hard a bump are we talking about and which car do you refer to? If it's only a slight bump then each car will probably only have damage to its bumper (fender in the US). But a harder smack may damage the radiator shell or even the radiator itself on the rear car and the boot (trunk) of the front car. A very hard smack might even damage the chassis of one or both cars.