An option is a derivative product that derives its value from the underlying asset, in case of stock market an option derives its value from stocks. So if a person is bullish about the stock or market then he or she will go for call option and if his or her view is bearish then he or she will go for put option. Option whether it is a call or a put option can be either an American style option or European style option.
While American option may be exercised anytime before the expiration date of option, so for example if option is of 31st December 2010 then American option can be exercised any time before 31st December 2010. Whereas European option can be exercised only on the expiration date of option, in case of above example if it is a European option then it can be exercised only on 31st December and not before that. In other words American style option is more flexible than European style option as it can be exercised anytime unlike European option which can be exercised only at particular date. Another difference between the two is that value of European option can be calculated using binomial pricing model because its expiration date is fixed whereas it is not easy to calculate value of American option as its expiration date is not fixed, it is dependent on investor or trader will.