What could cause the pain and possible swelling on my jaw joint?

Usually TMD. Usually a problem with the TMJ. See a TMJ expert to help evaluate and treat your problem. A TMJ expert can be general dentist, an oral surgeon, prosthodontist, orofacial pain specialist. Ask your physician, dentist or dental society for a recommendation.
TMJ. This could be caused by a dysfunction of your joint, known as TMJ syndrome. In this, there has been some injury to the jaw joint causing the pain and swelling. Overuse of the jaw, as in chewing gum too often, may cause such pain. Basically, the muscles which move the jaw begin to spasm, or cramp, and this causes the pain. The best treatment is to rest the jaw as much as possible.
Mechanical issues. It depends on the mechanics for the jaw joint. It could be mechanical or inflammatory. Analysis of the TMJ by imaging, examination can help clarify the problem.
Difficult issue. Temporomandibular joint pathology is a complicated and difficult issue and consulting a dentist is warranted.