Fungi are eukaryotic organisms that have chitin in their cell walls and are heterotrophs, they don't photosynthesize. Fungi include yeast, molds, and mushrooms.
If we compare fungi with plants, the obvious difference would be that plants photosynthesize and this means they have chlorophyll, so they are green and require light to grow. Another difference is that plants have cellulose in their cell walls.
The white button mushroom is the more common one, and probably the one found in the pizza. This is a member of the Basidiomycota, and the part that we consume is the fruiting body, although most of the time fungi are in the form of mycelium, which is formed by filaments called hyphae, where the cells are separated by septa.