Tendons are tough bands of connective tissues that attach a muscle to a bone. In general, they are not extremely stretchy as they are used to transmit forces — when muscle contraction happens, it takes up the slack in the tendon, and then the bone moves. The tendon allows the muscle to be biomechanically more efficient by changing where the leverage comes from. However, some tendons, like the Achilles, have elastic properties, allowing them to store and recover energy at high efficiency when loaded, like a spring.
Tendons are subject to many types of injuries including rupture and avulsions, which is where the tendon rips from the bone. The most accurate term for most overuse tendon injuries, however, is tendinopathy — which generally results in low grade, chronic irritation, and degeneration or weakening of the tendons, which may eventually lead to tendon rupture. ‘Tendinitis’ is actually less common and is the correct term for an abrupt inflammation of the tendon-like if you spend all weekend painting the house, and then suddenly had elbow pain.
A ligament attaches bone to bone and provides joint stability. It prevents excessive motion and maintains structure. They have a better ability to stretch and then regain their shape than a tendon, but when overstretched past a certain point or for an extended period of time, they are less likely to return to their original length. This is why ankle sprains or shoulder dislocations are best treated quickly and appropriately.
Ligament injury usually occurs from trauma or a fall when high force stresses the joint beyond its normal range. While ligaments are extremely strong, they can be stretched or even torn completely. Different grades of sprains depend on how far the tissue is stretched and dictates how quickly it heals (and if requires immobilization and/or surgical repair.)