Definition: An option contract is an agreement between two parties to buy/sell an asset (stock or futures contract as an example) at a fixed price and fixed date in the future.
It is called an option because the buyer is not obliged to carry out the transaction. If, over the life of the contract, the asset value decreases, the buyer can simply elect not to exercise his/her right to buy/sell the asset.
There are two types of option contracts - Call options and Put options. A Call option gives the buyer the right to buy the underlying asset, while a Put option gives the buyer the right to sell the underlying asset.
A simple example: Peter buys a Call option contract from Sarah. The contract states that Peter will buy 100 Microsoft shares from Sarah on the 5th May for $25. The current share price for Microsoft is $30.
Note: this is an example of a Call option as it gives Peter the right to buy the underlying asset.
If the share price of Microsoft is trading above $25 on the 5th May, then Peter will exercise the option and Sarah will have to sell him Microsoft shares for $25. With Microsoft trading anywhere above $25 Peter can make an instant profit by taking the shares from Sarah at the agreed price of $25 and then selling the shares on the open market for whatever the current share price is and making a profit.
The $25 value, which is stated in the agreement, is referred to as the Exercise (or Strike) Price. This is the price at which the asset will be exchanged.
The date (in this case 5th May) is known as the Expiry (or Maturity) Date. This date is the deadline for the option contract. At this date, the option buyer is to decide if a transaction of the underlying asset is to occur.
Outcomes: Let's imagine that at the expiration date, Microsoft is trading at $30, then Peter will buy the shares from Sarah at the agreed $25 and then he can sell them back on the open market for $30 and make an instant $5.
Alternatively, if Microsoft is trading at $20, then buying the shares from Sarah at $25 is too expensive as he can buy them on the open market for $20 and save $5. In this situation, Peter would choose not to exercise his right to buy the shares and let the options contract expire worthless. His only loss would be the amount that he paid to Sarah when he bought the contract, which is called the Option Premium - more on that a little later. Sarah would, however, keep the option premium received from Peter as her profit.
In the real world of exchange traded options, transactions don't really take place between two people like I've explained above. The process of Novation actually removes the identity of who is on the other side of the trade. You simply Buy or Sell an option contract from the exchange without knowing who is on the other side.