How to Play 9 Ball - The Basics
The Object of the Game
Simple enough. You play with 9 balls (numbers 1-9). After the break, you have to clear the balls in numerical order. The person to sink the nine ball wins.
Rack 'em in a diamond shape with the 1 ball at the top on the foot and the 9 ball in the center. The other balls don't have a specific order, so random is fine. Rack 'em tight.
Usually the one who wins the opening lag gets the first break. After the opening break, breaking order varies based on the rules you set up at the start. The most common breaking orders are either alternating or loser breaks. Be sure to set the break rules BEFORE the start of a game.
As for the break itself, you have to hit the 1 ball first for the break to be legal. If you sink a ball, you keep going. If you don't sink one, your turn is over. If you scratch, your opponent has ball in hand anywhere on the table.
Winning the Game
All you need to do is be the one who sinks the 9 ball. Going in order, the nine ball is going to be the last ball on the table. You can also win by sinking the nine on a combo as long as the first ball you hit is the lowest number ball on the table.
Those are the basics. Of course there are a lot of other rules dealing with push outs, fouls, bad hits, masse shots, jump shots and such. You should check out the BCA's official rules for more information on those topics.