Basidia are specialized end-cells on which the fungi that possess them produce their spores. The picture here shows various stages of spore-production on the basidia of Pisolithus tinctorius. In this case, you can see the spores becoming brown as they mature.
This series of drawings shows the development of this type of basidium.
A third type of phragmobasidium is simply a string of four cells, each bearing its own sterigma and spore. Unfortunately, I don't have a picture of this type of basidium yet. This type of basidium is typical of the order Auriculariales and the classes Ustilaginales (the smuts) and Uredinales (the rusts).