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Nearly all applications of Fuzzy logic rely on the notion of linguistic variables. These are variables whose values are words rather than cold hard numbers. Something like "it is nice outside" is an examples of a linquistic variable. These are values which don't necessarily directly relate to cold hard numbers, but, they do in a roundabout way. When I say that it is nice outside, that is subjective to my opinion; other people may have different opinions on what is considered nice outside. That is why this is called fuzzy logic: each fuzzy set carries some tolerance for imprecision. The tolerance for ambiguity helps us model the world in a more realistic form by using language rather than cold hard numbers. With words we can quickly convey ideas like "young" and "old" and quickly make actions based on this knowledge. Since there is no definitive answer on what is the cut of for being old/young, we can use fuzzy logic to deal with partial truth values.
Fuzzy Sets
Classical sets are mutually exclusive. In other words: things can only belong to one set at a time. In a fuzzy set, elements can belong to multiple sets with some degree of membership. As an example, someone who is 30 may be 33% in the young set and 66% in the old set. Fuzzy sets are usually are represented by trapezoids; however, other shapes such as gaussian can be used.