Being double-jointed really refers to an excess amount of motion in the joints. There aren't really two joints. The joints look the same as in the everyday, average person.
Another term for this condition is hypermobility. For example, some hypermobile people can bend their thumbs backwards to their wrists. Others can bend their knee joints backwards, or put their leg behind the head.
Most people with hypermobility have this in all the joints. This is referred to as generalized joint laxity. The basic difference between people with normal joints and hypermobile joints is that joint laxity allows hypermobile joints to stretch farther than is normal.
Although it may seem that having loose ligaments would prevent injuries, excessive joint laxity actually increases the risk of ligament rupture. This is especially true for the anterior cruciate ligament (ACL) of the knee. Without strong muscles on either side of the joint, loose joints have less stability. There is less tension to hold them from sliding too far in one direction and tearing or rupturing.
It seems that joint stiffness is a natural part of the aging process. Many people mention stiffness as they get older. And in fact, joint laxity decreases with age as well. This suggests a similar process in those who were seemingly double-jointed.