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Acromioclavicular, Coracohumeral, Coracoclavicular, and Glenohumeral ligaments
These ligaments are more-or-less to hold the shoulder together while all of the moving parts are moving. The collarbone, shoulder blades, and humerus are all moving around, so keeping them together is a large priority. When these ligaments aren't in tact, injuries occur at the shoulder joint.
The labrum is a structure that exists to increase the stability of the shoulder. The shoulder can move in a bunch of different directions, and that means the big ball on the humerus (the head of the humerus) will move around. Without making a pocket for that head to move around in, it would slide out of place, or dislocate.
Biceps Brachii Tendon and Triceps Brachii Tendon
There are other tendons in the arm, but for the sake of simplicity, they will not be listed on this site. Every muscle in the arm must have an attachment to a bone, and that attachment takes the form of a tendon. These two are listed here because they are the tendons of the most contributing muscles in the arm. These are paired tendons and muscles, meaning as one works, the other works to counter it for control of force and direction. These are also the most prevalent sources of injury according to the American Journal of Sports Medicine.
Ulnar Collateral Ligament (UCL)
The ulnar collateral ligament is the ligament that is lateral to the body (far away from) when you throw a ball. The ligament connects the humerus (the bone in the upper arm) and the ulna (the forearm bone on the pinky side). The main job of this ligament is to prevent internal translation of the elbow (the elbow moving towards the pinky without the shoulder or wrist moving) and prevent excessive external rotation, or the motion of showing your palm for a high five with your elbow parallel with your shoulder. This is involved in throwing, since in the cocking phase of the throw, the ligament is active in springing the arm forward. When this tendon is torn, the dreaded "Tommy John surgery" is required.
Radial Collateral Ligament (RCL) and Annular Ligament
The radial collateral ligament is attached on the thumb side of the elbow when throwing a ball. This is opposite on the elbow to the UCL. This ligament prevents against outward translation of the elbow and internal rotation (the opposite as the UCL). The annular ligament's main purpose is to keep the radius touching the humerus. There is less stress placed on these ligaments in a throwing motion. (via innerbody.com)
Wrist Flexor Tendons
Wrist flexor tendons are responsible for transferring the force from the muscles on the palm side of the forearm to the hand. The wrist will stay in an extended position (showing the heel of your hand), so this is responsible for pulling the fingers toward the palm and squeezing the ball.
Wrist Extensor Tendons
The wrist extensor tendons have the opposing purpose of the flexor tendons. They work to extend the fingers and extend the wrist. When the wrist is extended, the grip strength of the hand is much higher than the grip strength of the wrist when it is flexed down.
The retinaculum is a ligament which keeps the ligaments of the wrist bound together.