A contract saying someone can buy something at a predetermined price or on a specified future date. They don't have to buy it, but they can if they want to.
Options contracts can be pretty tricky to explain. Have you heard of a Lease Option for a house? It means you'll rent the house, but you'll also have the option to buy it. You literally sign a contract that says "I'll buy this for $X before X date, if I want to." You'll pay a little bit to get that contract because the owner won't be able to sell it to anyone while you still have the right to buy it.
You can write an option contract for anything, though. "I will buy this carrot for $2 before December 12, if I want to." Okay, so the contract uses a little more complicated language, but that's the idea. You see, you've gotten the option to buy it if you want. Because they lose the ability to sell it to anyone else, the owner will usually require a fee.
Now to add to that, people can buy and sell these contracts. They become little goods themselves. Their value is determined by the value of the object it's for, the price listed on the contract, and how close the expiration date is.
The Options market has become it's own financial market writing, trading, and claiming options contracts. Usually, when people talk about "options" (for short), they're talking about options contracts for stock shares. Options contracts also exist for commodities and currencies, though.