A bidet is generally defined as some sort of basin used to wash the genital area.
The word itself is French, meaning "small horse", and rhymes with "filet". It originally described the basin-like pieces in bathrooms/restrooms/water closets that were specifically designed for cleaning your nether regions with water. Water should be gently propelled toward the area so it covers and washes the outside without too much force as to possibly douche in women (which isn't generally advised). They're often shorter than toilets when they come as a separate unit and can contain one faucet or a hose with a shower-like head. While common in Europe, the Middle East, Asia and Latin American both in homes and hotels they're somewhat scarce in America. By those unfamiliar with them they're mistaken as foot washers, small toilets and places to do hand laundry. In new American homes and hotels they're more common since they are preferable to toilet paper and promote awareness and better hygiene.
Why should I use a bidet?
Bidets are better at cleaning than toilet paper of any variety, despite what the ads may claim. Even plain paper without any additional lotions or the like, which are promoted as soothing and comfortable, can be irritating to your genital area and aren't advised for use. The primary reason that they aren't common in America is as a whole Americans aren't comfortable being aware of and often touching their genitalia, especially after urinating or the like, and the point of these devices is to take better care of them and keep them cleaner. Using a bidet is often more comfortable for people with a variety of ailments such as hemorrhoids and unlike toilet paper it doesn't help spread or irritate the genitals when infected. They've been shown to help prevent vaginitis, yeast infections, UTIs and are preferred for people who have trouble bending and reaching to clean themselves.
Um, so how do I sit on this supposedly useful fixture?
The bidet earned the name "small horse" from French Cavalry men in the 15th century since they wanted some way of cleaning the area that was always in contact with their saddles and "straddling the pony" is both a slang way of referring to and a guide for sitting on these fixtures. A bidet doesn't have a seat so part of you may rest in the basin while facing the opposite direction you would on a toilet. Some bidets automatically spray water up at you and just require you to dry off afterward while others require you to fill the basin and just bathe that area. You can often find bidets built into toilets these days which solves the problem of moving from one fixture to another in a bathroom and saves space. They're generally activated by a button and simply spray water up at you, like the Japanese toilets you hear about in the West.
Sources:
Personal Experience
Go Ask Alice. Columbia University Health Services. 3 June 2004 <http://www.goaskalice.columbia.edu/>