A european option may be exercised by only at the maturity of the option, i.e. at a single point in time
An american option on the other hand may be exercised at any time before expiry. Oddly, american options are very rarely exercised. That is because any option has a time value and is therefore worth more unexercised. Buyers who wish to realise the full value of their option will therefore prefer sell it on a recognised market, either an over the counter (OTC) or listed market.
European options are typically valued using the Black-Scholes or Black 76 formulas. American options are more difficult to value, and a choice of models are available (for example Whaley, binomial options model, Monte Carlo and others).
Some other styles of options are