Basidiomycetes is a division of true fungi which are more commonly known as club fungi. They get their common name from the fact that they produce their spores at the tips of swollen hyphae that look a bit like clubs or baseball bats. These swollen tips are called basidia.
This division includes mushrooms, puffballs, earth stars, stinkhorns, bracket fungi, rusts, smuts, jelly fungi and bird's-nest fungi. These fungi generally reproduce sexually with the aid their spores, which are released from the hyphae in the gills of a basidiocarp (the "cap" of a mushroom is the basidiocarp).
From the BioTech Dictionary at http://biotech.icmb.utexas.edu/. For further information see the BioTech homenode.