A medical procedure where they surgically alter the genitals of a person to as closely resemble those of the opposite sex as much as possible.
It's an expensive procedure, ranging in the $8000-$20,000 range (depending on where you go) for the average surgery for male to female, and $15,000-$50,000 for female to male, and can run even more expensive depending on what needs to be done and the type of procedure used (different procedures yield different results). Any respectable surgeon requires letters of approval from both a psychologist and a psychiatrist who have had extended relationships with the transsexual desiring surgery.
Female to male surgery lags far behind male to female in terms of results. Male to female surgery, at its best, has been known to fool a gynecologist (I've heard of one story where a transsexual woman married her gynecologist, who never even knew).
Also sometimes called gender reassignment surgery.
The male to female procedure is called a vaginoplasty. First, an incision is made along the penis, from the base of the scrotum to the tip. Scrotal skin and the testicles are removed, and the skin of the penis is carefully separated from the erectile tissue and removed. The erectile tissue is then removed. In the meantime, the scrotal skin is reshaped into a closed tube which is grafted to the penile skin, to increase the depth of the vagina to be created, and the glans of the penis is reshaped and sewn into position to form the clitoris.
The skin of the lower abdomen is carefully lifted away, and the process of creating the vagina begins. The penile skin, with the attached graft, is inverted, and placed inside the cavity made by lifting away the skin. The penile skin is then carefully sewn into place. The urethra is also rerouted at this time. The remaining scrotal skin is trimmed to make the labia. Packing is placed into the vagina to help hold the shape during initial healing, and final stitching and cleaning up is done.
The female to male procedure is called a phalloplasty. The one described here is a free flap phalloplasty. First, a graft from the forearm is taken as the materials to work with - this includes skin, nerves, arteries, and veins. The skin is wrapped around a catheter, and constructed into a penile shape.
The vaginal cavity is obstructed (though I don't see a description how anywhere), and the labia majora are used to construct a scrotum. The clitoral nerves are routed into the base of the new penile tissue, and will grow into the tissue.
Often, a second surgery six to nine months later is performed, to add implants for the appearance of testicles, and to allow erection on demand.
See http://www.annelawrence.com/meltzersrs01.html for a series of images documenting the process. WARNING: Not for the squeamish, as they are quite graphic!
About the penis transplant listed in the writeup below:
The likelihood of this being true seems slim to me, considering that sex reassignment surgery, as I described above, does not involve simply amputating the penis, but remaking it. I suppose it would be possible to consider a transplant of the erectile tissue so that partial reconstruction could have been done on the patient, but the skin with the nerves and blood vessels is kept and used in the creation of the vagina. I guess it would be possible that the doctors there were doing a completely different kind of SRS, but not one that I am aware exists.