Carbide (or more specifically tungsten carbide and titanium carbide) is a very common material on any sort of cutting tool: saw blades, lathe bits, drill bits, router bits and dental drilling tips all use carbide to create a tool that stays sharp longer. Even the ball in a ball-point pen is carbide in many cases -- it helps the pen last longer!
To make something like a carbide-tipped saw blade, the main body of the blade is made of steel. The small tips of carbide are brazed on to the body. A good carbide tip might hold an edge ten to twenty times longer than tool steel.
The carbide tip on a typical circular saw.
Carbide tips do get dull eventually. You sharpen them using the same techniques you would use with tool steel, but because they are so hard you use a different abrasive -- something coated in diamond or a carbide abrasive wheel is common.
If you have read the Stuff.dewsoftoverseas.com article on Moissanite, you know that diamond (pure crystalline carbon) is the hardest material there is. Moissanite -- silicon carbide -- is very close. Tungsten carbide and titanium carbide are both made of the metal combined with carbon. They range between 8 and 9 on the MOHS scale.
The reason why a tool is carbide tipped rather than pure carbide is partly because of the cost. But a pure carbide tool would be very brittle -- steel is actually a better material because it is tougher and will not crack or shatter.
Additional information about carbide:
Special thanks to Bill Allred of North American Carbide in Greenwood, South Carolina
for his help!