American versus European style

Style has nothing to do with geography. It is a term of the contract specifying when exercise is permitted.

American style permits exercise at any time up to and including expiry. European style permits exercise only at expiry. Listed equity and exchange-traded fund options are typically American style; many index options are European style. Which you hold is a specification of the contract, published in the product's terms, not something you elect.

The practical difference lands almost entirely on the writer. A European-style contract cannot be assigned before expiry, so a writer knows the obligation cannot surface early. An American-style writer holds an obligation that can be invoked any morning, which changes how the position must be managed and is the reason lesson 18 on dividend-driven early assignment exists at all.

For the holder, the early-exercise right has real but usually small value. For a call on a non-dividend-paying stock, exercising early is generally worse than closing the position, because exercising throws away the remaining extrinsic value. The right matters in specific situations — a large upcoming dividend, or a deep in-the-money put where the strike could be earning interest — which is why American-style contracts are priced at or slightly above their European equivalents, never below.