Your best bet is to find a field guide and go through it. See our guide to identifying birds . What color are the legs. I've got a hawk in my back yard eating my birds. If your yard or the surrounding area is wooded, it's most likely a Cooper's Hawk.
Small birds are their main diet.
These hawks are as natural a part of the landscape as are the songbirds. By feeding the small birds, you're providing the hawk with a delightful buffet. The only thing you can do is stop feeding the birds until the Cooper's Hawk moves on. During migration, the hawk in question might be a Sharp-shinned Hawk. The Cooper's Hawk used to be a pretty rare bird.