Suppose on 1 March 2015 you buy an option to purchase 1 million barrel of oil for $80 per barrel on 1 July 2015. If on 1 July 2015 the market price of oil is $78 per barrel you definitely will not use your option, and will purchase oil on market price. But, if on 1 July 2015 the market price of oil is $82 per barrel, you will use your option, and will purchase 1 million barrel of oil for $80 per barrel.

So, when you have assumed a long position in a call option (you have option to buy) you use your option when market price S_T is greater than the strike price K, otherwise you do not use your option.

Hence, payoff from a long position in a call option is given by

Similarly, when you have assumed a long position in a put option (you have option to sell) you use your option when market price S_T is less than the strike price K, otherwise you do not use your option.

Hence, payoff from a long position in a put option is given by

A short position in a call option means you have sold a call option, so the buyer of the call will earn when market price is greater than the strike, and you will loose. So, the payoff from a short position in a call option is given by

Similarly, the payoff from a short position in a put option is given by

Both American and European options have same payoff except that the American options can be exercised at any time up to the expiry, while European options can only be exercised on expiry.