They are called ‘red-bellied’ because their belly is in fact red, although usually a bit hard to see. They do have red on the top of their head, and sometimes you’ll hear someone call it a Red-headed Woodpecker. Think of the Red-headed Woodpecker as the ‘Red-hooded’ Woodpecker, and maybe that will help. Both are on the Nature Trail.
If you look at a range map for the Red-bellied Woodpecker, you’ll see that it’s primarily a southeastern bird .
Red-bellied woodpeckers are noisy birds, and have many varied calls. Males tend to call and drum more frequently than females, but both sexes call. Often, these woodpeckers “drum” to attract mates. They tap on aluminum roofs, metal guttering, hollow trees and even transformer boxes, in urban environments, to communicate with potential partners. 
These birds mainly search out arthropods on tree trunks. They may also catch insects in flight. They are omnivores, eating insects, fruits, nuts and seeds. Their breeding habitat is usually deciduous forests.
They nest in the decayed cavities of dead trees (snags), old stumps, or in live trees that have softer wood such as elms, maples, or willows; both sexes assist in digging nesting cavities. The video below is of a pair of Red-bellied Woodpeckers nesting in a snag on the Nature Trail.
References and Additional Information
 Audubon: Red-bellied Woodpecker
 Wikipedia: Red-headed Woodpecker
 Cornell – All About Birds: Red-bellied Woodpecker > Sounds