(Taken from Wikipedia) slightly edited)

The pileus is the technical name for the cap, or cap-like part of a mushroom or fungus, of a basidiocarp or ascocarp (fungal fruiting body) that supports a spore-bearing surface, the hymenium.
The hymenium (hymenophore) may consist of lamellae, tubes, or teeth, on the underside of the pileus. A pileus is characteristic of agarics, boletes, some polypores, tooth fungi, and some ascomycetes.

Pilei can be formed in various shapes, and the shapes can change over the course of the developmental cycle of a fungus. The most familiar pileus shape is hemispherical or convex. Convex pilei often continue to expand as they mature until they become flat. Many well-known species have a convex pileus, including the button mushroom

Some, such as the parasol mushroom, have distinct bosses or umbos and are described as umbonate. An umbo is a knobby protrusion at the center of the cap. Some fungi, such as chanterelles have a funnel- or trumpet-shaped appearance. In these cases the pileus is termed infundibuliform.