If your mystery bird is tiny and round, with a grayish back, black cap, white cheeks, and a black chin, you've got yourself a chickadee. In the northern states, it would be a Black-capped Chickadee. From Philadelphia southward, it would be the look-alike Carolina Chickadee. Out west, it might be a Mountain Chickadee or Chestnut-backed Chickadee. They are all very common yard birds.
Now, if your bird is tiny and almost all gray, with a small crest and a blot of black above its squat beak, a Tufted Titmouse has stopped by your suet. A familiar backyard bird in the eastern U.S., the Titmouse can also be identified by the rusty wash under its wings.
A bit larger than the titmouse and chickadee is the White-breasted Nuthatch. This bird has a white-to-gray breast, slate back, and black on its cap and nape. Also notable: their long, thin beaks and tendency to scale trees and hang from feeders while upside down.
Speaking of the Hairy Woodpecker, it is probably the second most common woodpecker to visit feeders after the smaller Downy. The two birds also look very similar. To decide which you have, consider the beak and the size of the bird in question. As a general rule, if you have a classic suet cage and the bird is roughly the same length or smaller, you have a Downy. If the bird is a bit longer than the cage and has a thicker bill, it's a Hairy. As with the Downy, male Hairy Woodpeckers have red on the back of their heads.
Because of its distinguishing bright red cap, the Red-bellied Woodpecker is sometimes confused with the Red-headed Woodpecker. But when compared side by side, they are very different looking birds. Red-heads are also not nearly as common. An abundant woodpecker in the eastern U.S., the Red-belly is a medium-size woodpecker. In addition to its red cap, it has a black-and-white striped back and a wash of red on its creamy-gray belly, which is how the bird got its name.
One of the more striking birds you'll see around your backyard, the Northern Flicker is a large woodpecker that often visits suet feeders. With their spotted breasts, striped backs, black bib, and a flash of yellow or red in the wings when they fly, they are stunning birds that look like nothing else hanging from your suet feeder. Enjoy.