What are long and short positions, and what are their payoffs?

Long Position
The party that agrees to buy the underlying asset in a forward contract is said to assumes long position. And, the payoff can be calculated as

Short Position
The party that agrees to sell the underlying asset in a forward contract is said to assumes short position. And, the payoff can be calculated as

For example, on 1 March 2015 party A agrees to buy 10 million pounds from party B at an exchange rate of 1.5 dollars per pound on 1 October 2015. The party B agreed with the contract. In this contract party A assumes a long position, and the party B assumes a short position. And, if on 1 October 2015 the exchange rate is 1.55 dollars per pound, then the payoff for party A is:

So, the party A gains $0.5 million in this example.

And, the payoff of party B is calculated as

The negative sign shows that the party B losses $0.5 million. Hence, the gain obtain by one party is the loss incur by the other party.