Before we discuss what it means to be "double-jointed" (a term we'll analyze closer in a moment), we should learn a little about how a "normal" joint works.
A joint is basically where two bones meet, allowing one to move against the other. The ends of the bones are tipped with cartilage where contact is made to prevent damage. Ligaments and other connective tissue hold the bones together. The motion is caused by muscle contraction or extension, and the muscles are attached to the bones by tendons.
Most joints allow for a standard range of motion. For instance, your elbow allows you to bend your arm and straighten it. If your arm was extended past the point where it essentially formed a straight line, it would likely cause a dislocation of the joint -- a painful separation of the bones and the ligaments that hold them in place.
However, some people do have a larger range of motion in their joints than others. The term double-jointed is commonly used, but it's not accurate. Try joint hypermobility or joint hyperlaxity instead. A person with hypermobility in the elbow may be able to extend his or her arm 10 degrees or so beyond what most of us consider to be a full extension.
How is this possible? Genetics play a large role, because the shape of the ligaments and the bone structure in large part determines the amount of motion a joint will have. Your hips and shoulders both have ball-and-socket joints: The end of one bone has a bulb that fits into a cuplike space on the other bone. If the ball is deep in the socket, the range of motion will be quite good, but not as good as when the ball rests shallowly in the socket. In fact, some people can roll the ball out of the socket and then roll it back in.