As I was walking past a tree in Fayetteville, I noticed some white objects about the size of small acorns. I stopped and knelt down to examine them, and it turned out that they were mushrooms!
Earthstars in the genus Geastrum were huddled around the base of the tree. The white sphere you see is the storage structure for the mushroom's spores. When force is applied to it (such as from a gust of wind or a finger), the spores puff out in a cloud from a hole at the tip. The spores are then dispersed by the wind. The brown stuff you see under the sphere is the "skin" of the mushroom (outer peridium): it peels back from the sack of spores and lies on the ground.
Mushrooms like these are relatively common once you know what to look for. Often, the earthstars and related groups look like small piles of decaying leaves or poop unless you take a closer look. If you see something gross on the ground, stop and take a closer look, it might be a mushroom.
EDIT: February 22 - Adding some photos of three collected specimens.