"Conure" is, in this case, a word representing a vaguely defined group of birds in the parrot order, psittaciformes. The term is usually used in aviculture rather than actual scientific ornithology as a way to differentiate bird breeds by size. Conures range in size from large parakeets (which in itself is a word used scientifically as a vague way of referring to even smaller parrot subspecies, or to Budgies if you're an American) to small true parrots. A lot of times people will actually interchange the word "conure" with "parakeet" since both groups of birds are so ill defined (again, unless you're an American, where the word parakeet is specifically referring to budgies). The reason for the word's misuse is that it was originally used to refer to the now-defunct genus conurus-- a genus that originally held some of the genera listed below.
Species of conures usually range from 10-12 inches in length (making them roughly the size of cockatiels). All conure breeds originate in South and Central America, save for the now extinct Carolina Conure which used to live in North America, but died out in the 1900's. They are tropical and neotropical, and all have the typical parrot curved beak and four toes.
While details vary species to species, conures in general are highly social, playful, inquisitive, and vocal. Many species are kept as companion birds.
While the term "conure" in aviculture is vague, it does specifically include several genera: Pyrrhura, Aratinga, Cyanoliseus, Enicognathus, Guarouba, Leptosittaca, Ognorhynchus, and the Conuropsis whose only member is the previously-mentioned extinct Carolina Conure. Most of these genera have single species, but Aratinga and Pyrrhura both have multiple species.
Some of the most popular pet conure species are the:
Sun Conure
Jenday Conure
Nanday Conure
Green-Cheeked Conure
Blue Crowned Conure
Patagonian Conure
Gold-capped Conure
Cherry Headed Conure
Maroon Bellied Conure
Some conures of interest include the:
Golden-Plumed Conure
Golden Conure
Yellow-Eared Conure
There are many many many more than this, but given the hodge-podge state of species classification for these birds, to even attempt to list them all would be a fool's errand as even now some species are being reevaluated.