Craniopagus parasiticus, or "attached parasitic head"* is an extremely rare birth defect similar to conjoined--"Siamese"--twins, except with only one person.

Let me try and explain:

Craniopagus twins are conjoined twins attached at the head. The recently deceased Nida and Hira Jamal of Pakistan were craniopagus twins. Two bodies, two heads.

Now, individuals with craniopagus parasiticus have two heads, but only one body. On top of their own head is another, mostly undeveloped head of a twin that did not grow into a full body. This head may have its own brain, eyes, lips, ears, and other assorted head goodies. While breastfeeding, the second mouth may move as well as the first.

On February 7, 2004, the first ever surgery to remove a "parasite head" took place in the Dominican Republic. Seven-week old Rebeca Martinez underwent a grueling surgery to remove the head of her undeveloped twin from her own. Of particular concern to the doctors was the fact that the second head was growing faster than the first, which would damage Rebeca's brain and make it almost impossible for her to lift her head by the time she was three months old.

The surgery was, unfortunately, unsuccessful, and Rebeca died twelve hours afterwards. This was not entirely unexpected, as the heads shared major arteries that needed to be severed.

Out of the eight documented cases of craniopagus parasiticus, Rebeca was the only to make it to birth: the rest died prenatally, making this surgery the first of its kind. Sadly, she was not able to pull through.

* I don't actually speak Latin, so it might mean something else altogether. When someone in the know informs me of a more exact translation, I will update this node.

Sources:
http://www.msnbc.msn.com/id/4194938/
http://everything2.com/index.pl?node_id=1143405