belong to the order of Passeniformes (perching birds) and the family of Fringillidae (true-finches). They are seed-eating song birds which explains why they are so common at bird feeders. There are four sub-families, 23 genera, and 994 species (from Wikipedia).
Those subsets shown elsewhere are highlighted in green.

Family Fringillidae
Subfamily Fringillinaefringilline finches. Three species, which feed their young on insects and few if any seed
Genus Fringilla
Chaffinch Fringilla coelebs
Blue Chaffinch Fringilla teydea
Brambling Fringilla montifringilla
Subfamily Carduelinaecardueline finches. A much larger group that contains several genera which feed their young mainly on seeds. This subfamily consists of several well-defined clades.
Typical grosbeaks - see Grosbeaks
Bullfinches - see Bullfinches
Genus Pinicola – Pine Grosbeak
Pyrrhula – bullfinches
Arid-zone clade
Genus Leucosticte – mountain finches
Genus N.N. –
Dark-breasted Rosefinch, "Carpodacus" nipalensis
RhodopechysTrumpeter Finch and relatives
American rosefinches
Genus Carpodacus
Cassin's Finch, Carpodacus cassinii
Purple Finch, Carpodacus purpureus
House Finch, Carpodacus mexicanus
Asian rosefinches
Genus Carpodacus – rosefinches – large number
Haematospiza – Scarlet Finch
Uragus – streaked rosefinches
Goldfinch-canary-crossbill clade (Carduelis and Serinus are polyphyletic and are probably best regarded as a collection of independent sub-groups or genera.)
    • Yellow-fronted Canary Serinus mozambicus
    • (Sub)Genus Crithagra – Afrotropical canaries, African seedeaters, citrils (probably only distantly related to Serinus)
  • Genus Linurgus – Oriole Finch (possibly basal to Crithagra)
  • Genus Rhynchostruthus – golden-winged grosbeaks (tentatively placed here)
  • Genus "Serinus" thibetanusTibetan Serin (probably related to Spinus and Serinus sensu stricto)
  • Genus Carduelis sensu lato
  • Greenfinch or European Greenfinch Carduelis chloris
    • (Sub)Genus Acanthis – redpolls (related to Loxia)
    • (Sub)Genus Loxia – crossbills
Carduelinae incertae sedis
Genus Pyrrhoplectes – Gold-naped Finch
Callacanthis – Spectacled Finch
Neospiza – Sao Tomé Grosbeak
Subfamily Drepanidinae – see Hawaiian honeycreepers.
Subfamily EuphoniinaeEuphonious finches - see Weaver Finches.
Genus Euphonia – euphonias
Genus Chlorophonia – chlorophonias


Family Thraupidae
Genus Geospiza endemic to the Galapagos Islands
Large Cactus Finch (Geospiza conirostris)
Sharp-beaked Ground Finch (Geospiza difficilis)
Vampire Finch (Geospiza difficilis septentrionalis)
Medium Ground Finch (Geospiza fortis)
Small Ground Finch (Geospiza fuliginosa)
Large Ground Finch (Geospiza magnirostris)
Common Cactus Finch (Geospiza scandens)
Genus Diuca
Genus Incaspiza 5 Inca Finch
Great Inca Finch, Incaspiza pulchra