How are transsexual operations performed?

Simply put, the person's own tissue is used to create appropriate genitalia for them.

Female-to-male -- This is usually done in three stages. First there is a bilateral mastectomy. Then the complete female reproductive system is removed. Lastly, some sort of phallus is created. The simplest way to do that is to clip the clitoral hood to make the clitoris, likely enlarged via hormone use, protrude and resemble a tiny penis. The vagina is occluded and artificial testes may be inserted. In a more complex surgery to create a phallus, skin grafts are used to build a tube, and tissue and vessels from the non-dominant arm are used as well.

Male-to-female -- In the most common surgery of this type, the testes are excised and the outer penile skin is inverted and moved to the new location. The inner tissue is mostly removed except for enough to form a clitoris. If the penile skin is inadequate, ancillary grafts or rectal tissue can be used. Surgeons use the scrotum to form the labia.