The four fungus classifications are Chytridiomycota (chytrids), Zygomycota (bread moulds), Ascomycota (yeasts and sac fungi), and Basidiomycota (basidiomycetes) (club fungi).
The form and internal structure of the sporangia that produce the spores are the most crucial features for distinguishing these four major groups.
Deuteromycota (fungi imperfecti) and lichens are two more conventional taxonomic groupings that aren't officially recognized (i.e., they are polyphyletic).
All fungi that have lost their ability to reproduce sexually belong to the Deuteromycota.
Lichens, unlike other fungi, are a symbiotic relationship between a fungus and an alga rather than a single organism. Without the algal symbiont, the fungal partner is usually unable to flourish, making categorization difficult.
They'll be regarded as a different category here, but it's important to remember that they're not single creatures or monophyletic groups.
Slime moulds and water moulds are examples of Oomycota.
Slime moulds are now understood to be a composite of three or four distinct taxa, and oomycetes are now classed with diatoms and brown algae in the Chromista.