Let me preface by saying the vast majority of fractures will heal (by osteogenesis or formation of new bone) without any need for surgery if the broken bone is properly aligned ("bone setting"). This occurs with varying degrees of union and loss of mobility as a complication--from full recovery of normal range of motion to total loss of limb function. For example, the collar bone has a remarkable ability to heal completely without any treatment, while a pelvic fracture would likely have resulted in permanent disability if the victim survived the internal hemorrhage (bleeding). Furthermore, children usually suffer 'green stick' fractures (which means the bone isn't broken but bends), so they don't often require surgery.

The standard method for the healing of fractures however was immobilization (restriction of activity of the limb) and bed rest. Ancient Hindus treated fractures with bamboo splints, and the writings of Hippocrates discuss management of fractures in some detail, recommending wooden splints plus exercise to prevent muscle atrophy during the immobilization. In fact, the method he described for reducing a dislocated shoulder is almost the same way it is done today. The ancient Greeks also used waxes and resins to create stiffened bandages, while Celsus describes how to use splints and bandages stiffened with starch. Arabian doctors used lime derived from sea shells and albumen from egg whites to stiffen bandages. All these were the precursors of the various casts that are used today.

Even though external fixation (in which holes are drilled into uninjured areas of bones around the fracture and special bolts or wires are screwed into the holes) is considered to be a rather "new" trend in orthopedics and traumatology, in fact it has been something used by physicians and surgeons for thousands of years. Hippocrates also described a form of external fixation to splint a fracture of the tibia. The device consisted of closely fitting proximal and distal Egyptian leather rings connected by four wooden rods.

The amputations that were carried out in the American Civil War (and wars before that) were a medical necessity in the pre-antibiotic era because such fractures were almost always compound or open fractures and the wounds "dirty" (meaning anything from dirt to pieces of clothing in contact with the wound, usually driven in by a projectile into the wound). Dirty wounds are prone to infection, which could be deadly if the limb became gangrenous.

Open Reduction Internal Fixation techniques are often used today in cases involving serious fractures such as comminuted (i.e fragmented) or displaced fractures or in cases where the bone would otherwise not heal correctly with casting or splinting alone. Open reduction refers to open surgery to set bones, as well as debride the wound. Internal fixation refers to fixation of screws and/or plates, intramedullary bone nails (femur, tibia, humerus) to enable or facilitate healing.

Background: physician

I'd just like to leave this here. I am by no means an expert in the area, but I find this surgical papyrus from Ancient Egypt amazing. The site explains it better than I would. Here's a print-screen from a passage on broken upper arm.

EDIT: If you'd like some more information about ancient surgical procedures, there's also the roman medical treatises in De Medicina of Celsus. Specially the books VII (ch. V on extracting weapons) and VIII (about fractures, luxations, etc.). Here's the chapter on fractures of the limbs.

I hope it helps.