The radiator support or core support is the frontmost piece of structure in an auto body. Its first name was radiator support, so-called because the radiator is attached to it, but in reality it is the most important part of the front end group. It controls the alignment of the fenders and the hood, and ties all of the parts together - even the headlights are attached to the core support, as are the horns.
In full frame or stub frame vehicles with a body or at least the front end group sitting atop a chassis or frame, the front end group is called the doghouse and the core support is a free-floating member of this structure. The core support is usually attached to the chassis itself (usually at the frame horns) with two or four rubber bushings. Attached to this are the fenders and fender skirts which provide the majority of the front end. The hood latch is typically part of the core support, though it is sometimes separate, and in rare occasions the hood opens forward, so the hood hinges are mounted here, or to both the fenders and core support. Forward-opening hoods are much more common on unibody vehicles.
On unibody vehicles the core support is welded in, and it connects to the inner fenders and the lower front unibody, specifically the integral frame rails. Full-frame cars use the fenders for structure, but unibody cars use the inner fenders, which are also welded into the entire body of the vehicle. Otherwise it is the same as on full-frame cars. Replacement requires cutting out the old and welding in a new radiator support.