Two brake parts get confused because they bolt together: the caliper and the caliper bracket. They do different jobs, and knowing which is which helps at service time. Here is the difference.
The caliper
The brake caliper is the hydraulic clamp - it houses the piston(s) that, when you press the pedal, squeeze the brake pads against the rotor. On a floating caliper it also slides on pins to center itself. The caliper is the moving, pressure-applying part.
The caliper bracket
The caliper bracket (or carrier/anchor) is the fixed piece that bolts to the steering knuckle and holds the pads in position and the caliper over the rotor; see the caliper bracket. It does not move or apply pressure - it is the anchor that transfers braking force to the vehicle. The slide pins ride in the bracket.
How they work together
The bracket holds the pads and provides the rails they slide on; the caliper mounts to the bracket (via slide pins on a floating design) and does the clamping. The pads sit in the bracket; the caliper squeezes them. Both must be clean and free-moving for even braking - rust in the bracket causes pad rattle and sticking.
Why it matters at service
At a brake job you unbolt the caliper from the bracket to change pads, and often remove the bracket to change the rotor. The bracket's bolts are torque-critical. Clean both, grease the pins and pad contacts, and fit fresh hardware with quality pads.