If done correctly, strength training results in just that, increased strength in your muscles, as well as in your tendons. For some people, increased tendon strength benefits them on many levels, while others may wish to keep their tendons loose and flexible. The ways to achieve this and reasons for doing so are entirely relative to that person and the specific tendons in question.
Function
Muscles don't attach to muscles, and they certainly don't attach to bones. Muscles attach to tendons, which are made up of mostly collagen. Tendons are stretched tightly and attach muscles to the bone. Every muscle responsible for moving a bone in your body is attached to the bone via a tendon. If a tendon tears or ruptures, it may make it impossible to move that bone until the tendon is repaired.
Misconceptions
The stronger a tendon is, the thicker it is, which in turn makes it tighter and less flexible. When a tendon is stretched out and limber, that tendon is weaker than a tight tendon. The weak, bendable aspect makes the tendon extremely difficult to tear, which is a positive thing for those who wish to remain as flexible as possible and aren't worried about sports-related strength.
Benefits
A high amount of stress occurs when a strong tendon is stretched out; however, when it returns to form, the recoil action is extremely strong. This is especially beneficial for athletes. Shoulder and elbow tendons that are strong and tight give quarterbacks and baseball players a competitive edge over their opponents, allowing them to sling a ball with more velocity due to the amount of stress in their recoil motion. Sprinters want strong tendons in their knees to help apply as much force as possible when their feet hit the ground.
Considerations
Even athletes don't want all of their tendons to be strong and tight. For instance, a runner wants his knee and ankle tendons to be tight to allow for maximum thrust. At the same time, he'll work to keep his hip tendons as loose and flexible as possible through long, specialized stretching sessions. If you're aiming to increase tendon strength, make sure you're doing so with tendons that you want to be strong and tight, not loose and flexible.
Warning
Tendon strengthening exercises often involve lifting more than your one repetition maximum but doing so in short, controlled movements. It's extremely important to use a spotter during these exercises, as one misstep can do some serious damage. A torn tendon may require surgery followed by months of rehabilitation with a physical trainer.
