Briefly: a kind of teacup. Less briefly, a kind of teacup distinguished (indeed, even in Chinese: the name literally means »lid cup«) by having three, not two, components: the cup, the saucer, and a lid, which is used to cover the cup (helping the heat retention of the tea) and strain the liquor: by tilting the lid very slightly, the drinker can prevent fugitive leaves from escaping the cup into his mouth while drinking. As this implies, tea is typically steeped directly in the gaiwan rather than poured into it from a pot, with fresh water poured on as the drink ebbs; for various reasons this is regarded as preferable by many enthusiasts.
The saucer often has a hole in the middle, again unlike the typical Western teacup design; the base of the cup fits into this aperture with some precision, keeping the whole assembly together when lifted — the classical way of drinking from a gaiwan is to raise the bowl to the lips by lifting the saucer with one hand, and to manipulate the lid with the other.
The gaiwan may be used in gongfu tea brewing in place of a teapot; in this case, a large amount of tea is used relative to the volume of the cup. The tea is steeped in the gaiwan, then poured from it into a number of smaller cups after a very short steeping (it may be as little as ten seconds); successful pouring requires a particular deftness of hand which, to paraphrase Musashi, takes careful practice. A well-made Yixing teapot is considerably easier to deal with in this respect.