The name is a joke that stuck: Bermuda sits between America and Europe, and so does the exercise right. The holder may exercise on specific dates — quarterly, say — but not continuously.
You will not find Bermudan contracts on a retail options-chain, but they are everywhere in institutional markets: callable bonds, cancellable swaps and many structured-product wrappers embed one.
Example: a five-year note is callable by the issuer each anniversary. The issuer holds a Bermudan call on its own debt with four exercise dates. The coupon you receive is higher than a non-callable note precisely because you sold that option, whether or not the term sheet uses the word.
Related: european-style-option, american-style-option, convertible-bond