The term "Fuzzy Logic" is a sort-of catch all way of describing a system that has more than two states.
"Fuzzy Logic" is different from "Classical Logic" because in classical logic there are some basic 'rules' that Aristotle laid out for systems of deduction. One of these rules is that there can be only two answers to the question "What is the value of this input/sentence/argument?", and those answers are True or False (in a computer context "1" or "0"). Fuzzy logic rejects that there can be only two answers to this question, creating either a third value - usually described as "both" or "neither" True and/or False - or creating a wide range of values that can be used in the valuation of an input.
Think of a computer with an attached mouse. The "click" function of the mouse is a classic logic-style situation because either you press the pad to input or you do not press the pad for an input. The value is either True - the pad is pressed and there is an input, or False - the pad is not pressed and there is no input. However, the movement of the mouse as it relates to the position of the cursor on the screen has many more inputs than just True or False. We may be able to describe the concept of "moving" or "not moving" in a classical way, but the actual position of the cursor has as many values as there are positions on the screen.
Another way of thinking about Fuzzy Logic is that it is not as clear-cut as "Yes" or "No" like classical logic. The more values you have to add to a system, the more complicated our understanding of the relationship between the input and the outcome under the more complicated situation. We can't just say that the input happened, we have to account for the surrounding context of the input. This makes the systems less straight-forward, and therefore "fuzzy".