Fungi are usually classified in four divisions: the Chytridiomycota (chytrids), Zygomycota (bread molds), Ascomycota (yeasts and sac fungi), and the Basidiomycota (club fungi). Placement into a division is based on the way in which the fungus reproduces sexually. The shape and internal structure of the sporangia, which produce the spores, are the most useful character for identifying these various major groups.
There are also two conventional groups which are not recognized as formal taxonomic groups (ie. they are polyphyletic); these are the Deuteromycota (fungi imperfecti), and the lichens. The Deuteromycota includes all fungi which have lost the ability to reproduce sexually. As a result, it is not known for certain into which group they should be placed, and thus the Deuteromycota becomes a convenient place to dump them until someone gets around to working out their biology.
Unlike other fungi, the lichens are not a single organism, but rather a symbiotic association between a fungus and an alga. The fungal member of the lichen is usually an ascomycete or basidiomycete, and the alga is usually a cyanobacterium or a chlorophyte (green alga). Often the fungal partner is unable to grow without the algal symbiont, making it difficult to classify these organisms. They will be treated here as a separate group, but it should be realized that they are neither single organisms, nor a monophyletic group.
It should also be noted that some organisms carry the name of mold or fungus, but are NOT classified in the Kingdom Fungi. These include the slime molds and water molds (Oomycota). The slime molds are now known to be a mixture of three or four unrelated groups, and the oomycetes are now classified in the Chromista, with the diatoms and brown algae.
The Tree of Life maintains an excellent page on the systematics of fungi.