Car shudders when brakes applied

Sometimes it can be a combination of brakes and another problem in the suspension or wheel bearings.

Replace the brake pads and machine/replace the brake rotors and everything may be fine at first.
What can happen after a while is that the new rotors may warp a few thousandths of an inch. While you may not even normally notice this in most cases, if you have a loose wheel bearing or suspension part this small amount of warpage will be magnified and made to feel worse than it actually is.

The front suspension should be checked along with the brakes.