An option is a contract that involves a buyer and a seller, and like all contracts it provides rights and imposes some obligations. What's being bought or sold is the underlying instrument, the underlying interest, or the just plain underlying. For simplicity, let's stick to equity options contracts with the underlying being shares of stock. One option contract generally represents 100 underlying shares.
Let's start with some vocabulary. A person who buys options is also called a holder. On the other side of the contract, an option seller is called a writer (they "wrote" the contract and sold it to you). A call contract gives the holder the right to buy 100 shares at a certain price for a set amount of time, and the writer has to sell those shares at that price if the holder exercises. A put contract gives the holder the right to sell 100 shares at a certain price for a set amount of time, and the writer has to buy them. If you buy a contract, you will pay what the market thinks that contract is worth, and that's called the premium. The contract writer gets to keep that premium. The premium is quoted on a per-share basis, so with a contract representing 100 underlying shares, when you see an options quote for $1.00, you know it will cost you $100 to buy that contract.