In British English, "Would you fancy a chat?" or "Do you fancy a drink?" means "Would you like a chat/drink?" and is a invitation to share a chat or drinks with the speaker. (As Bernadette helpfully points out in her comment below, "fancy" in this context means "desire.") Even more informally, the "would/do you" is dropped and just assumed to be understood - "You fancy a chat/drink?" Americans generally do not use "fancy" as a verb, and would mostly just be puzzled by this question. But, as Clinton said, Americans might use fancy as an adjective and order "fancy drinks." These drinks might be made from several kinds of liquor, served in a special type of glass, and be decorated with things like cocktail umbrellas or slices of fruit.

Do you mean, "Fancy a chat?" or "Fancy a drink?" Are they the phrases you heard? "Fancy...?" as a verb means "would you like...?" It can also be a noun, or an adjective (as Clinton has described).

What does it mean? Not dull. Elaborate and exceptional from a social or ornamental point of view.

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