What is a "limit switch" for a central heating unit? And is it expensive to replace?

A limit switch is a switch that opens or closes at a certain temperature, the limit. Older furnaces used them to turn on the blower when the heat exchanger warmed up. Furnaces still use one to shut down the furnace if it over heats because the blower didn't start up or something.

I think they only run a few dollars. If you can identify the culprit, find it, and remove it, you may get away about as cheap as any repair other than a blown fuse. May have to invade an electrical or HVAC supply to buy one. Might try a hardware store first. They are usually friendlier to DIY. Often they are a round doodad with a diamond shaped bracket and 2 terminals. If they don't have a rating marked on them, you may have to take the model number of your furnace to a dealer for your brand of furnace.