They are small to moderately large and have a strong beak, usually conical and in some species very large. All have 12 tail feathers and 9 primaries. These birds have a bouncing flight with alternating bouts of flapping and gliding on closed wings, and most sing well. Their nests are basket-shaped and built in trees.
Species
Family Fringillidae
Subfamily Fringillinae - Fringilline finches; contains only three species, which feed their young insects rather than seeds.
There are many other birds in other groups which are called finches, notably the very similar-looking Estrildids or waxbills, which occur in the Old World tropics and Australia.
Some of the closely related sparrows are also named as "finches", as are some buntings.
Some other Cardueline finch species not listed above:
This proposal would move them to Fringillidae, as a subfamily (and therefore also force recognition of subfamily Carduelinae for the fringillids in South America).
Since that paper is based on just mitochondrial DNA, it certainly is not the last word, but it does suggest that Euphonia and Chlorophonia belong within Fringillidae proper.
Another problem is that based on Yuri and Mindell, Euphonia and Chlorophonia could cluster within Fringillidae, in which case we'd have to do away with Euphoniinae.