A "put" option allows the owner to sell 100 shares of a given stock at a given price, called the strike price. This right is good only for a certain period of time. American-style options allow the owner to do this (called "exercise" the option) at any time up to the time the option expires (called "expiry" or "expiration date"). The owner can do this, but is not required to do this.
This would only be done if the price of the shares has fallen below the strike price.
For example, suppose you own one option to sell XYZ at $60 on or before the third Friday of November. If the price of XYZ is $44 on that date, you could exercise the option and receive $6,000 by selling those shares. This protects you by letting you sell the shares above the current share price. For this chance (after all, the stock could also be trading at $74 on that date, in which case you wouldn't exercise the option), you paid somebody a premium.
There are two people for each contract. For a put, the buyer wants to sell the 100 shares for each contract bought (option contracts are almost always in blocks of 100 shares) at the strike price. The buyer would exercise the put, sell the shares (going short the shares), then buy them back at the lower price, pocketing the difference. In this instance, the buyer does not end up owning the shares at the end. The seller is paid the premium, which he or she keeps regardless of what happens subsequently. The seller is betting (literally) that the share price will be above the strike price by the expiry, at which point the profit is the premium received. If it is, he or she won't have to buy the shares because the buyer could just sell those shares at the higher price.
The owner of the put option is said to be "long" the put.
The buyer is betting that the stock price will go down before the expiry date, while the seller is either betting that it will not or is content to buy the shares at the strike price.