I would look at three things in this order:
The caliper bolts (on which the caliper slides in and out). They can get rusty or dry, and cause the caliper to stick.
The can be cleaned and lubed.
The caliper base, which bolts to the spindle. There are sliding surfaces on which the moving part of the caliper rides. They can be cleaned up if rusty, but must be replaced if badly worn or eaten away.
The caliper piston. If it's rusted/stuck, you're probably best off replacing the whole caliper.