Selling shares that you don't own is known as short selling. You'll do this when you think the market proposition won't happen and want to take a position that's opposite to that stated in the proposition. You're making the opposite trade to those buying the shares from you, who say the market proposition will happen.

Short selling is simple to do - you just sell the shares (see How do I buy and sell shares?). If you want to exit your position at a later time you'll need to buy back the shares that you sold. For example, if you short sell 10 shares then you must buy back those 10 shares with another trade to exit your position.

Let's look at this in the context of a trade...

There is a proposition listed for the Saints to beat the Rams. Believing the Saints will not beat the Rams (remember that you always short sell shares when you believe the market proposition won't happen) you short sell 25 shares to a buyer. You now have a position of -25 shares for the Saints to win (the negative number indicates you have a short position on this proposition). The buyer you traded with now has a position of +25 shares for the Saints to win.

As you can see, you didn't need to own any shares before short selling them - you simply enter an order to sell the shares. To exit your position later you buy back the shares that you sold with a new trade.

To make a profit short selling you need to sell high and then buy back for a lower price. By buying low and selling high, you'll make a profit - but in this case, you've done the selling part of the equation first.