Concise list of bra types and definitions/defining features purpose, along with images to demonstrate shape and design. Although this is not the entire list of bra models and makes, this list covers the vast majority and qualities held by the styles.
(Definitions sourced from the Wikipedia 'List of brassieres' page...because I live dangerously...)
The adhesive bra, sometimes called a "stick-on bra" or a pair of "pasties", adheres to the breasts, often with one piece for each breast. There are no straps or bands of any sort. Provides little support to the breasts. Usually intended for backless and strapless fashions. There are two types: disposable paper bras that use a strong adhesive, and re-usable ones that use silicone and can be washed and re-used.
The balconette bra is a design that lifts the woman's breasts. First designed in the United States in about 1938, and came into mainstream fashion in the 1950s.
The bullet bra is a type of full-support bra with cups in the shape of a paraboloid of revolution with its axis perpendicular to the breast. The bullet bra usually features concentric circles of decorative stitching centred on the nipples. Invented in the late-1940s, they were still being sold well into the 1980s.
The convertible bra has straps that may be detached and rearranged in different ways depending on the outfit. Alternatives to regular straps for strapless bras are beaded bra-straps or clear plastic bra-straps that provide support to the woman's bosom.
The cupless bra, also called an open cup bra, or sometimes a shelf bra (see Shelf bra below), consist of a bra frame with no breast support or limited cups that only support a small portion of the lower half of the breast. Usually worn as erotic lingerie, a cupless bra can cause the shape of the nipples to be prominently visible beneath an outer garment.
The demibra, also known as balconette bras, are a half-cup bra style with wide-set straps and a horizontal bust line. Often being seamed or boned, these promote cleavage and a bra-like effect. These offer less support but enable low-cut garments to be worn without the woman's bra being seen. Demibras can be designed to provide lift as do push-up bras.
The front-closure bra: a bras that has closures, usually hooks and eyes, on its front rather than at its back. These bras lie almost completely flat at the woman's back. Some of these bras feature a racerback-style back (see below).
The full support bra is a type of bra that is designed to offer good support for the entire breasts and, this is a typical, practical bra for everyday wear by a full-busted woman.
The mastectomy bra is designed such that a breast prostheses is held in place, and these are intended for women who have lost one or both breasts by either mastectomy surgery for breast cancer or women who have been severely injured in a breast, such as by a gunshot wound or a traffic accident.
The maternity bra is designed such that its cups do expand to adjust as the breasts' increase in size over the course of a pregnancy. The term "maternity bra" can also refer to a nursing bra.
The minimizer bra is designed to de-emphasize the woman's bosom, in particular that of a large-breasted woman (size 32C and above). The minimizer bra, by compressing and shaping both breasts, helps to create the impression that a woman has a smaller bosom.
The padded bra is simply a bra with cloth or plastic foam padding inside its lining, around, over, or below the woman's breast. This bra is designed to give a fuller shape for a small-bosomed woman. Of course, this bra is an alternative to stuffing the bra with paper tissues, socks, cotton balls, sugar packets, plastic foam, etc. Unlike push-up bras, most padded bras support the woman's breasts, but they do not significantly lift them.
The plunge bra is generally worn with a low-cut blouse, dress, or sweater. (This can also be the design of a bikini top). As suggested by its name, this bra plunges down between the woman's two breasts. This allows the bra to remain covered by even a low neckline. In general, a bra that is exposed under a low-cut shirt worn in public is considered to be unfashionable and possibly even mildly distasteful by many people. In addition to remaining invisible under low necklines, the plunge bra will create and accentuate the line of the breasts.
The push-up bra is structured so that the woman's breasts are lifted and placed closer together for cleavage enhancement by the woman. During the last few decades the best-known brand of push-up bra is the Wonderbra. Many push-up bras contain padding, typically made of foam or rubber, but some contain gel-filled pads. The main distinction between padded bras and push-up bras that incorporate padding is whether the padding is centered under the breasts to lift them or is outside the center so that the padding pushes the breasts inward.
The racerback bra has shoulder straps that come over the shoulder in a "V" pattern very close to the neck. This design is sometimes worn under dresses or tops where "traditional" straps would be exposed. Many sports bras use racerback construction because this design is effective in providing good support to the breasts so that they do not bounce around very much.
The sports bra provides firm support for both breasts, it is meant to prevent discomfort and stretching/sagging of the breasts during exercise times. Some women prefer to wear a sports bras exclusively, or even while sleeping.
The strapless bra, which has no shoulder straps, is designed for women wearing clothing that reveals her shoulders, collarbone area, and perhaps the top halves of the breasts—as far low as the tops of the areola. This bra is good for wearing beneath strapless evening gowns by women who choose not to go braless.
Training bras are designed for girls who have begun to develop breasts but for whom a standard-size bra does not fit properly. They are of simple construction and offer little, if any, support, but may aid in concealing developing nipples and breast buds and helping young girls grow used to putting on and wearing brassieres.
The T-shirt bra is designed without any raised seams, hooks, etc., so that they fit against the wearer's skin smoothly, allowing the wearer to don a tight T-shirt, sweater, or sweatshirt with minimal bra visibility. These bras sometimes include a small amount of padding in their cups in order to conceal the nipples.
Underwire bras are designed to give extra support and endurance. They have a wire (metal or plastic) running under each cup to provide rigid support to the bust.
*Very interesting research, discovering bra types I wasn't even aware of (U-plunge, for example)...and I call myself a Brassiere fanatic! And yes, I did feel like a smutty teenage boy spending a good hour finding pictures of women in bras.