Answer:
A call option gives its buyer the right, but not the obligation, to purchase (or "call in," which is where the name comes from) something from the person who sold him the option at a specified price.
A put option gives its buyer the right, but not the obligation, to sell (or put in the other person's hands) something to the person who sold him the option, once again at a specified price.
In both cases, the person buying the option has to pay money, which is called a premium.
Let's look at this from the buyer's standpoint. You trade stocks to make money. If you think a stock is going to go up a lot, you can buy a call. If the stock price exceeds the price on the call (aka "strike price") you exercise the call--you buy the stock from the call's seller--then sell the stock on the open market and pocket the difference.
On the other hand, if you think a stock is going to go down in price, you buy a put. (A good investment strategy here is to buy a "naked put"--one where you don't own the stock you're writing against yet.) If the stock DOES go down in price, you exercise the put. The money for the stock will wind up in your trading account. You've got 72 hours from that point to buy enough stock to fulfill the contract if you don't already have it, so you take the money you were paid--you get that right away--have your broker get you the stock you need, turn it over to the put seller and pocket the profit.
Both these options have expiration dates. If a put expires with the stock priced above the strike price or a call expires with the stock priced below the strike price, the option "expires worthless." If you bought a put at $20 and the stock trades at $25 on the expiration date, you'd be really dumb to go through with the trade...so no one does it.