It is the removal of an infected toe nail, but there is no pain involved because the doctor deadens the feeling in the toe first. See the related link for a video of the process. CAUTION: It is graphic!
Ungiectomy is the surgical removal of part or all of a toenail, often to treat ingrown toenails. Matrixectomy involves removing part of the nail bed to prevent the nail from regrowing in cases of chronic ingrown toenails. Both procedures are typically performed under local anesthesia.