We have a standard ugly fluorescent overhead fixture in the kitchen. This morning the light just went off. I went to see if the bulb had burned out and the light switch itself was hot.

What could cause this? I had not noticed the switch ever getting hot before. It wasn't really super hot but enough heat to cause concern.

The switch is currently not working. So is it the bulb (burned out) or did the switch blow and I need to replace that one?

I think only the switch is defective, the contacts have been worn and this caused the heat you feel. One fixture is not enough to cause an overload, but the current of one fixture certainly can heat up worn contacts.

Simply replace the switch with a new one, and you should be all set.