Basidiomycota

Basidiomycota is a group of higher fungi. It is one of two large phyla that make up the subkingdom Dikarya. The other is the Ascomycota. 
These are "higher fungi". They includes jelly and shelf, or bracket fungi, mushrooms, puffballs, and stinkhorns; and the rusts and smuts.
Agaricomycotina.
This sub-phylum includes the "classic" mushrooms, polypores, corals, chanterelles, crusts, puffballs and stinkhorns. The three classes in the Agaricomycotina are the Agaricomycetes, the Dacrymycetes, and the Tremellomycetes.
Pucciniomycotina.
The Pucciniomycotina include the rust fungi, the insect parasitic/symbiotic genus "Septobasidium", a former group of smut fungi (in the Microbotryomycetes, which includes mirror yeasts), and a mixture of odd, infrequently seen, or seldom recognized fungi, often parasitic on plants.
Ustilaginomycotina.
The Ustilaginomycotina are most (but not all) of the former smut fungi and the Exobasidiales. The classes of the Ustilaginomycotina are the Exobasidiomycetes, the Entorrhizomycetes, and the Ustilaginomycetes.