This is an extremely complex topic, on which numerous books have been written. The basic concept however is fairly simple.

An option is a contract that gives you (the owner of the option) the right to buy or sell a stock (or other financial instrument, such as an ETF, etc.) at a specified price, which could be higher or lower than its current price, within a specified period of time.

Once the specified period of time has passed and you did not exercise your right to buy or sell the financial product in question, the option expires and is worthless. A "call" is an option to buy, and a "put" is an option to sell the underlying financial product. You can also write an option, or sell an option you bought, as long as it is not expired.

You might be puzzled at how it is possible for an option to allow you to buy a stock below its current value, or sell it above its current value. The answer is that the option itself is sold at a premium, which is higher than the difference between its "strike price" (i.e. the price at which you may buy or sell according to the contract) and the price of the underlying financial product at the time you purchased the option.

The next question might be why it makes sense to buy an option instead of simply buying the underlying stock. The answer to that is that if the price of the stock in question is say $50 per share, you'd need to spend $50,000 to buy 1000 shares, and have the ability to make $1000 if the stock price goes up by $1 per share. If you buy an option to buy 1000 shares at $50 and the current price is $50, the option will be much cheaper than $50,000, and if the share price goes up to $51, you can exercise your option, buy the 1000 shares at $50 a piece, and sell them at the new current price of $51, making an instant $1000 gain. Alternatively, and more simply, you can sell your option for just under $1000, since that is its value at that time.

So, you may ask, why does anyone buy stocks instead of options? The answer is that options are highly leveraged. If you buy the underlying stock and it drops by 1%, you lose 1% of your investment. However, if you buy an option to buy that stock at its current price (or higher) and the stock price drops even 0.001% and stays there until your option expires, you will lose 100% of your investment. This is why so many books have been written about options trading. It is a great way to make or lose a lot of money very quickly, and obviously you prefer the making of money over its loss.
Source(s):
http://www.optionseducation.org/basics/whatis/

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