Options represent the right (but not the obligation) to take some sort of action by a predetermined date. That right is the buying or selling of shares of the underlying stock.

A call is the option to buy the underlying stock at a predetermined price (the strike price) by a predetermined date (the expiry).
You buy a call if you think the price is going up. You can get it at a discount.
If the price goes down you don't do anything.
You Sell a Call to make money if the price won't go up more than the fee you get for selling the call

A put is the option to sell the underlying stock at a predetermined strike price until a fixed expiry date.
You buy a put to hedge against a decrease in price of a stock.
The seller is betting the price won't drop and they can keep the fee.
A put (which you bought and can be sold) increases in value as the underlying stock decreases in value.
You sell a put to make some extra money, usually if you hold shares of blue chip stocks which you think won't go down.

Relatively few options actually expire and see shares change hands. Options are, after all, tradable securities. As circumstances change, investors can lock in their profits (or losses) by buying (or selling) an opposite option contract to their original action.

Links:
Basics of Options Trading | Call Options | Put Options at the Motley Fool

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