The scientific name for a red-headed woodpecker is Melanerpes erythrocephalus.

You're not providing much information to go on, like behavior, size, habitat, etc, but you might want to look in a field guide and see if Hose Finch (Carpodacus mexicanus) is what you are asking about.

Cats are a predator of the Red-browed Finch. The Red-browed Finch is commonly found in grassy areas of a backyard.

house finch, gold finch, purple finch, zebra finch, parrot finch, society finch, bengalese finch, lady gouldian finch, spice finch, red bow firetail finch, canary finch, Australian finch

Red-headed Lovebird was created in 1758.