The term basidia refers to the finger-like projections found on the underside of mushrooms and other fungi. At the ends of these projections lie the reproductive spores. When the spores are mature, a mechanism within the fungi forces the spores to detach themselves and shoot off into the air. Scientists aren't sure what causes the separation, but have observed tiny droplets of fluid at the points of detachment, leading them to suppose it might be chemical in nature.
In comparison, an ascus is a transparent bag that forms in several different fungi species In some asci, spore cells divide within the bag and multiply through a dual process of meiosis/mitosis to produce one or more spores lined up in a single chain, or arranged in multiples of two or four like peas in a pod. At maturity, the bags rupture, spewing the spores outward by the hundreds into a little cloud that resembles talcum powder. In other species, there's a lid that opens to release the spores; still others have no active way to disperse their spores at all. (These have bags that dissolve in water allowing the spores to float away.)
Though vastly different, a basidia and an ascus both serve a very important function in the dispersal of spores from fungi.