The term arthritis refers to an irregularity within a joint, with weardown of the cartilage that is needed to cushion the joint.  When this cartilage, or "cushion," wears down, there is bare bone which articulates at the joint, producing pain and inflammation.
There are several types of arthritis, basically grouped into two large categories, posttraumatic or degenerative, and inflammatory.

Posttraumatic Arthritis and Degenerative Arthritis
Posttraumatic arthritis results from either an injury or a fracture that went into a joint, and degenerative arthritis is a "wear and tear" arthritis in which the joint for some reason wears down due to abnormal use.
Any joint within the hand or wrist can become arthritic due to an injury.  Following such an injury, posttraumatic arthritis often takes several years to develop.  Sometimes an individual will not even recall the injury, and develops an arthritis of unknown initial cause.
Arthritis within the hand joints can be treated with silastic or plastic prostheses.  Whether such a prosthesis is indicated or appropriate depends on the site of the arthritis, the stability of the joint, and the physical requirements of the patient.
Probably the most common type of degenerative arthritis is carpometacarpal (CMC) arthritis.

Inflammatory Arthritis, or Carpometacarpal Arthritis (CMC Arthritis)
The second major category of arthritis, inflammatory arthritis, is part of a disease process that causes inflammation of the joint, which ultimately proceeds to destroy the cartilage.  One well-known example of inflammatory arthritis is rheumatoid arthritis, but there are many others,.
Many examples of inflammatory arthritis exist, including the arthritis that occurs with lupus erythematosis, psoriasis, and other disease processes. Probably the most well-known type is rheumatoid arthritis.