What are those bumps on my horse's jaw?

The bumps are eruption bumps. Usually. They are the bilateral enlargement of the mandible to accommodate the growth of the permanent tooth roots. The permanent tooth ceases growth after loss of the baby (deciduous) tooth and eruption the permanent tooth.

The bumps may be slightly painful to the horse if firm pressure is applied. The bone covering the roots is very thin and can be damaged easily; be careful with chain leads under the jaw.

The bumps slowly disappear after the permanent tooth erupts, but this may take several years. Horse with smaller heads usually have more prominent bumps because the mandible is not as deep as a horse with a large head. 

The area on the nose, lateral to the bridge of the nose, below the white line on the picture is also enlarging because of root development. Pressure on this area from a too tight nose band may cause head problems during training.