Quarter cup bras support the lower part of the bust and are ideally worn as a fashionable item as opposed to a functional/practical item. This is due to the lack of full bust support. They are not suitable for bigger and/or less firm breasts.
Strapless bras have removable straps making them suitable to be worn under strapless dresses or tops. A common issue with these bras however, is that they don't provide enough support for bigger (and therefore heavier) breasts.
Half cup bras (also known as demi cup) are designed to cover up to just above the nipple and offer less support than a full cup bra due to covering and holding less of the breast. These bras are suitable for low cut tops and dresses.
Balconette bras are lower at the front, making them suitable for lower cut tops and dresses. They are suitable for most sizes, big or small, and have been described to give a nice rounded shape to breasts.