There are two bones in the forearm: the radius (on the thumb side of the arm) and the ulna (on the little-finger side). The heads, or upper ends, of these two bones are part of the elbow, and are "connected" to the humerus (the upper-arm bone) by several ligaments, which are bands of tough, flexible connective tissue. One of these ligaments, called the annular ligament because of its ring-like shape, is anchored on the head of the ulna but wraps around the head of the radius. This ligament is also connected to the radius, but not as well as to the ulna.
When your arm and hand are pulled and twisted, you can tear a small part of the annular ligament. When your arm is released after the pull part of the ligament is pinched between the head of the radius and the end of the humerus. This hurts. It also makes it hard for you to move the elbow without more pain. Usually a child with nursemaid's elbow hold the arm still to avoid pain; they may hold the affected elbow straight or a little bent, with the radial side turned inward (we call this pronation). She will be pretty comfortable unless you try to move her elbow or to turn the forearm back outward (supination). There is usually no swelling, but the radial side of the elbow may be a little tender.
Although we call this subluxation (partial dislocation) of the radial head, the radial head is not actually out of place -- it's just that the ligament is trapped in the joint. X-rays of the elbow are usually normal, since the radial head isn't really dislocated, and I usually don't get X-rays of nursemaid's elbows unless I can't reduce them or unless I think there may be another injury beyond the radial head subluxation.
Reducing a nursemaid's elbow is actually fairly simple once you have been shown how to do it once or twice. (Good thing, too, since once a child has had one nursemaid's elbow there is a fair risk of having another subluxation in the same elbow.) When I reduce a nursemaid's elbow, I will often show parents how to reduce it themselves. In many cases the child's movement is enough to reduce the subluxation, or my moving the arm and elbow during examination reduces the subluxation without any other measures. You need to ask your own doctor about the procedure -- it's hard to describe over the Net without demonstrating. Rarely (I've never had to do it myself) a child's nursemaid's elbow cannot be reduced this way and an orthopedist has to take care of the problem; the chance of a fracture in a case like this is much higher, and X-rays are needed at this point to make sure that the radius and ulna are both intact.