The current stock price you're referring to is actually the price of the last trade. It is a historical price – but during market hours, that's usually mere seconds ago for very liquid stocks.
Whereas, the bid and ask are the best potential prices that buyers and sellers are willing to transact at: the bid for the buying side, and the ask for the selling side. But, think of the bid and ask prices you see as "tip of the iceberg" prices. That is:
The "Bid: 13.20 x200" is an indication that there are potential buyers bidding $13.20 for up to 200 shares. Their bids are the highest currently bid; and there are others in line behind with lower bid prices. So the "bid" you're seeing is actually the best bid price at that moment. If you entered a "market" order to sell more than 200 shares, part of your order would likely be filled at a lower price.
The "Ask: 13.27 x1,000" is an indication that there are potential sellers asking $13.27 for up to 1000 shares. Their ask prices are the lowest currently asked; and there are others in line behind with higher ask prices. So the "ask" you're seeing is the best asking price at that moment. If you entered a "market" order to buy more than 1000 shares, part of your order would likely be filled at a higher price.
A transaction takes place when either a potential buyer is willing to pay the asking price, or a potential seller is willing to accept the bid price, or else they meet in the middle if both buyers and sellers change their orders.
Note: There are primarily two kinds of stock exchanges. The one I just described is a typical order-driven matched bargain market, and perhaps the kind you're referring to.
The other kind is a quote-driven over-the-counter market where there is a market-maker, as JohnFx already mentioned. In those cases, the spread between the bid & ask goes to the market maker as compensation for making a market in a stock. For a liquid stock that is easy for the market maker to turn around and buy/sell to somebody else, the spread is small (narrow). For illiquid stocks that are harder to deal in, the spread is larger (wide) to compensate the market-maker having to potentially carry the stock in inventory for some period of time, during which there's a risk to him if it moves in the wrong direction.
Finally ... if you wanted to buy 1000 shares, you could enter a market order, in which case as described above you'll pay $13.27. If you wanted to buy your shares at no more than $13.22 instead, i.e. the so-called "current" price, then you would enter a limit order for 1000 shares at $13.22. And more to the point, your order would become the new highest-bid price (until somebody else accepts your bid for their shares.) Of course, there's no guarantee that with a limit order that you will get filled; your order could expire at the end of the day if nobody accepts your bid.