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.
For instance, chanterelles are usually (as in this picture) 5- to 7-sterigmate. Just be thankful we use the numbers, and there aren't a lot of extra prefixes to learn!
Here, the development of one type of phragmobasidium can be seen: it starts off, in the top left, as just an end cell with a swelling at the end, but splits into two, and then four cells, each with its own sterigma. This type of basidium is known as a tremelloid basidium, and is typical of the genus Tremella. Fungi with this type of genus are placed in the order Tremellales.
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).