What Are Possible Leg Injuries?

I would definitely take him to the vet......It could be an arthritis issue depending on the age of the cat it could also be something more serious that isn't allowing him to put any pressure on the limb.  It could have been bitten by something as well that is causing an irritation.

Limping can be due to trauma (fall, fight, hit by car, etc.) resulting in soft tissue/muscle injury or fracture of the long bones.  A sprain or strain can result is pain and discomfort.  Dislocation of the hip joint is another injury seen in cats.  
Fight wounds from cat fights or wild animal fights can be very painful, cause an inflammation of the soft tissue (cellulitis) and can abscess.  Abscesses are very common causes of lameness and discomfort in cats.
Foreign objects in the foot or other part of the leg will cause pain.  Grass awns, foxtails, or cactus are common depending upon your area of the country.
Arthritis is usually a slow-onset cause of discomfort or lameness.  They very rarely won't put weight on an arthritic joint suddenly.  In an older cat cancer can cause pain or discomfort.
It would be best to have your cat examined by a veterinarian to identify a cause, start treatment, and appropriate pain medications.

Broken leg, pulling a muscle, twisting an ankle and losing a leg.

I found an abandon kitten that was only about a couple of weeks old. I have been bottle feeding her with kitten formula and she has learned to use the litter box now that I have had her for about two and a half weeks.  She is frisky, but favors her front left paw.  When walking, she limps, but when I examine her paw she does not react as if in any pain and she uses both front paws in trying to climb.  She is fast in running, but still favors the front left paw when walking or running.  What could be wrong?