The true bullfinches are thick-billed finches in the family Fringillidae and genus Pyrrhula. These birds are restricted to the Old World, and are:
Eurasian Bullfinch, Pyrrhula pyrrhula
Brown Bullfinch, Pyrrhula nipalensis
White-cheeked Bullfinch, Pyrrhula leucogenis
Orange Bullfinch, Pyrrhula aurantiaca
Red-headed Bullfinch, Pyrrhula erythrocephala
Grey-headed Bullfinch, Pyrrhula erythaca
Azores Bullfinch, Pyrrhula murina

There is an unrelated group of New World birds also named as bullfinches because of their superficial resemblances to the above species. These passerines were placed in the large bunting and American "sparrow" family Emberizidae, but are now considered tanagers Thraupidae. Four are in the genus Loxigilla:
Puerto Rican Bullfinch, Loxigilla portoricensis
Greater Antillean Bullfinch, Loxigilla violacea
Lesser Antillean Bullfinch, Loxigilla noctis
Barbados Bullfinch, Loxigilla barbadensis

The fifth is
Cuban Bullfinch, Melopyrrha nigra