A European option is an option that can be exercised only on the expiry date of the option, i.e. at a single pre-defined point in time.
An American option is an option that can be exercised at any time before the expiry date.
Other options where early exercise occurs differently are given below. However note that the pay-off value remains the same as a standard option.
A Bermudan option is an option where the buyer has the right to exercise at a set number of times that is always discretely spaced. This is the via-media between a European option–which allows exercise at the expiry only–and an American option, which allows exercise at any time before expiry date. Bermuda geographically is situated between America and Europe and hence the name Bermudan option for this.
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