Check out the wikipedia article on fuzzy logic and the tutorial (links below).
Regular logic involves things that are either true or not true. For example: My towel is wet or my towel is not wet.
Fuzzy logic allows for degrees of truth. My towel may be partially wet, or 65% percent wet (some areas may be dry or the towel isn't totally saturated).
Regular logic would state that it's either cold or not cold. Fuzzy logic would allow for various degrees of cold. It could be cold but maybe not as cold as very cold. It's all very fuzzy (haha).
Regular logic is black and white. Fuzzy logic is more vague.
I don't really know much about fuzzy logic but I was interested in your question, so I decided to learn about it myself.
Here's a list of applications:
Fuzzy logic is used in the operation or programming of:
Air conditioners
Automobile and such vehicle subsystems as automatic transmissions, ABS and cruise control
Tokyo monorail
Cameras
Digital image processing, such as edge detection
Dishwashers
Elevators
Some microcontrollers and microprocessors (e.g. Freescale 68HC12)
Hydrometeor classification algorithms for polarimetric weather radar
Language filters on message boards and chat rooms for filtering out offensive text
The Massive engine used in the Lord of the Rings films, which allowed large-scale armies to enact random yet orderly movements
Mineral Deposit estimation
Pattern recognition in Remote Sensing
Rice cookers
Video game artificial intelligence
Home appliances (e.g. washing machine)