Any of the tendons around the wrist may become inflamed and tender.  Cause and treatment are respectively similar for the management of each of these tendons.
The most common form of tendonitis around the wrist is deQuervain's tenosynovitis.

Anatomy
There are six compartments at the wrist through which the tendons on the back of the wrist run.  The first compartment is on the thumb side of the wrist, and the two tendons that extend the thumb backwards and move the thumb away from the palm run through this compartment.

Symptoms
Pain on the thumb side of the wrist occurs when the tendons of the wrist, the extensor pollicis brevis and the abductor pollicis longus, become inflamed within the compartment due to swelling or irritation from repetitive thumb movements.  As the tendons move through the tunnel in this area, they become irritated such that lifting the thumb away from the palm, making a fist, and, in particular, moving the wrist toward the small finger become extremely painful.  These symptoms are associated with difficulty moving the thumb, and there may be a snapping or catching sensation as the thumb flexes and extends.
This condition generally develops over time but may present itself over a period of days.  It tends to worsen with repetitive use of the hand and thumb.

Diagnosis
A diagnosis of de Quervain's is made using a test called the Finkelstein test, in which the thumb is held in the palm, and the fingers cover the thumb.  The wrist is then moved toward the small-finger side, reproducing the pain by stretching the inflamed tendons.

Treatment
As with all cumulative-trauma problems, the most important line of treatment is to refrain from the activity that caused the problem in the first place.  If the injurious exposure is discontinued, with time the problem also will resolve.
Associated with this modification of activity should be other conservative measures, including oral antiinflammatory medications and splinting the thumb to provide rest for the afflicted part.
A cortisone injection into the first dorsal compartment also can reduce the inflammation in this local area and may be very effective in resolving the problem.  Injection of cortisone can cause atrophy of these tissues and some change in skin color and should therefore be undertaken with caution.  Cortisone injections given into other areas on the back of the hand should also be undertaken with particular care, as cortisone can weaken tendons or even effect tendon rupture.

Surgery
If conservative care does not improve the situation and the individual is unable to engage in daily activities, cannot work, and finds the problem interfering substantially with his or her ability to function, then surgical release of the first dorsal compartment at the wrist is indicated.
This is a simple procedure in which the compartment is divided, thus allowing the tendons to glide freely.
In order not to injure the sensitive superficial skin nerves in the area, this procedure should be done with great care and with the assistance of magnification, using specialized glasses or a microscope during surgery.

Healing after Surgery
Following surgery, a splint is applied.  Therapy is commenced to help the individual gradually return to normal activities.  Over a period of one to two months, special exercises are done both onsite and as part of a home program.