lith = stone (Latin from Greek, e.g. lithograph, neolithic)
otomy = cutting (Latin from the Greek thomos/temnein, e.g. microtome, dichotomy)
Lithotomy is the extraction of a calculus (or stone) from an organ -- historically from the urinary bladder.
The lithotomy position is when a person is supine with knees raised and spread. The etymology suggests that the position was developed for the practice of stone excision -- a practice dating back to at least Hippocrates (mentioned in his eponymous oath). One account tells of a French physician first suggesting it for birthing in 1668. Allegedly, le Roi, Louis XIV witnessed the birth of his mistresses' children while they were in this position and it started a fashion trend.
I'll expose my bias here by noting that while run-of-the-mill obstetricians may prefer delivery in this position because it helps to immobilize the mother and give easy visual and manual access to the doctors and nurses, most women -- when given the choice, choose not to labor and deliver in this position and midwives have never advocated it particularly. (In my experience, midwives are more concerned with the patient's comfort than their own convenience and as such suggest that the laboring mother adopt the position that makes her most comfortable -- which, I understand, is not usually lithotomy. Squatting, crouching, hands'n'knees, floating in a tub, lying on the side seem to be more common.)
The lithotomy position is commonly employed for the above mentioned purposes as well as gynecologic pelvic exams, a number of other surgical procedures that require access to the perineum as well as a range of practices involving mutilation of female genitalia.
The lithotomy position has been implicated as a cause of circulatory complications during extended surgeries and increased incidence and severity of episiotomy during child birth.