The splint bones are small bones on either side of the cannon that run from the knee to the ankle. These bones are kind of fragile and easily become inflamed from a blow or heavy work, especially in younger horses. Boots are sometimes used to protect the splint bone.
When we say a horse has a "splint," we mean that the horse has a bump on the splint bone. This is generally caused by either inflammation of the ligament between the splint bone and the cannon bone, or by a minor fracture to the splint bone itself. These injuries often heal with extra bone growth around the damage, causing an obvious bump. The horse is generally lame when the injury first occurs, but usually recovers sound. (Unless the splint bone is fractured, in which case surgery is sometimes performed). Treatment for a splint injury is generally to administer an NSAID (usually bute) and ice the area, and may also involve stall rest and/or the use of liniments. In some cases the bump may need to be surgically removed, for example if it's interfering with the suspensory ligament. However, many older horses "have splints" that don't affect them in any way.