Torn Ulnar Collateral Ligament
A torn ulnar collateral ligament is the most feared of all injuries as a pitcher, partially because of its prevalence (the most common season-ending injury for pitchers), and partially because of it's severity (healing time requires at least 8 months according to the American Society of Sports Medicine). The injury requires surgery, and if the tear is small enough, the two torn portions can be sewn together. If the tear is too great, another length of tendon or ligament is inserted into the injured area and the ligament is reconstructed. It is generally taken from a donor or possibly the other arm. More information available at about.com
Miami Marlins' Chris Volstad. Notice the scar on the inside portion of his elbow.
Torn Shoulder Ligaments
Whenever one of the ligaments in the shoulder get damaged, whether that be by trauma (direct contact injury), inflammation of another structure nearby, or overuse, the shoulder loses stability. When the shoulder loses stability, the moving parts that occupy the shoulder lose their ability to function properly, causing pain and loss of strength and accuracy. This is handled by surgery if it is serious enough. The shoulder can also dislocate with the tearing of the "glenoid labrum."
Information and image via the Ohio State University Wexner Medical Center
Torn Rotator Cuff
This can happen much the same way that torn ligaments in the shoulder happen. When the rotator cuff is torn, the stability of the shoulder is minimized. The humerus can become dislocated much easier, which causes pain and lack of strength and accuracy of the movements of the upper arm. More Information
Tendonitis is the inflammation of a tendon, and in pitchers, the most common site for tendonitis is the proximal biceps tendon (where the bicep attaches to bone in the shoulder) and the distal triceps tendon (where the triceps attach at the elbow). This is treated by using ice, rest, and antiinflammatory medications (such as ibuprofen). This generally does not require surgery, however it dos cause pain and may linger for a long duration of time.