What is the procedure used to set broken bones without surgery?
When surgery is not used, we call this a 'reduction'. Some fractures may need to undergo 'skin-traction' following the initial reduction.
The fractured bone, usually one of the bones in the arms or legs is manupilated by an orthopedic specialist whilst the patient is sedated. This is done so that the ends of the bones can be realigned; in which case, the fracture is said to be reduced. Once a successful reduction has been confirmed by x-ray, the fractured bone is splinted in a plaster cast so that the realigned bones remain in place, and the patient is usually free to go home so their fracture can heal. This of course depends on which bone it was that was fractured.
Some bones may require the application of skin-traction for a few days in order to monitor the progress of the bone's repair through a series of repeat x-ray imaging. In which case, a cable is anchored to a cast or brace, and then passed over a pulley and weighted on the other end with a suitably-sized weight whilst the patient remains bed-bound. The addition of weight adds an element of continuous force that counteracts the force of the muscles that attach to the fractured bone, preventing them from pulling the realigned bones back out of place and allowing the bone to heal properly.
Skeletal traction on the other hand requires the insertion of surgical pins or screws into the bone near to the fracture site so as to provide an anchoring point for the traction to be applied; this is usually required for bones that are under much more force and so need heavier weights i.e. femur fracture or neck fractures. However, there is a surgical element to those procedures, so it wouldn't be fitting of your question.