Fuzzy logic is a mathematical methodology (and a philosophical ideology) that is similar in construct to boolean algebra and similar in appearance to probability, but more general than both in fundamental ideas. This generality is the freedom for truth variables to hold any value between 0 and 1 (inclusive), and fuzzy logic proponents claim this generality allows greater flexibility, freedom, accuracy and compactness when representing real world situations. All the usual properties of boolean algebra can be extended to fuzzy logic, and probability's degree of belief in a boolean variable becomes a fuzzy variable's degree of truth.
Fuzzy logic has had a great deal of success where it has been applied in the real world, and is often touted as a means of making machines smarter. However, for the most part, fuzzy logic has actually been used in control systems as a way of providing more human-like behaviour. For example, many household appliances (notably white goods such as washing machines and refrigerators) now contain fuzzy control systems that allow the machine to adjust to the specific circumstances currently presented to it, with some even learning patterns of usage. While this does improve the machines and their behaviour, it isn't really an advance in the areas of general intelligence on arbitrary domains and machine learning. Only (relatively) recently is the area of fuzzy machine learning being developed, with the most popular approach being fuzzy neural networks (often abbreviated to simply neuro-fuzzy) to allow machine intelligences to learn from arbitrary empirical data and experiences.