Badminton is a racquet sport, played on a rectangular court with a net about as high as halfway between a volleyball net and a tennis net. The object of the game is to hit the "Shuttlecock" so that it lands on your opponent's side of the court. Of course, they're trying to hit it back to your side of the court. The shuttlecock is usually a piece of rubber, with something attached, usually a light plastic mesh in the shape of a cone, although sometimes it has feathers attached, designed in a way that when it travels through the air, the head of the shuttlecock is always pointing in the dirction the shuttlecock is moving.
How to play Badminton:
Get yourself a racquet. Badminton rackets are fairly small, compared to those used in other racquet sports. This makes sense, since the shuttlecock is lighter than a tennis or a squash ball. In particular, the shaft of the racquet looks a lot thinner than what most people would be used to. If someone tried using one to play tennis, I really wouldn't be surprised if it had broken by the end of the game.
Get some shuttlecocks. They're fairly cheap. Depending on where you're playing, you might lose a fair number of them, such as on the roof. If you're just starting out, it's really easy to hit the shuttlecock a lot further than you thought you were going to.
Find a court. If you can't access an actual Badminton Court, hijack a volleyball court, or just rig one together in the backyard, hanging up a net, or just a piece of rope at the proper height (1.55m) for the net. Anything over the rope counts, anything under doesn't.
Serving the shuttlecock. Man, that sounds dirty. The server stands within the "service court", about the back 2/3rds of the court. Holding the shuttlecock below the waist, they hit it towards the part of their opponent's court diagonally opposite from them. At the moment of impact, the racquet should be angled such that all parts of the head of the racquet is lower than the hand.
Once hit, it should fly over the net, so that it would land on your opponent's service court, diametrically opposed from the one you're in.
If you screw up the serve, it then becomes your opponent's turn to serve. You switch serving from the right to left sides of the court and vice versa after each of your serves.
Scoring. Only the side that is currently serving can score. They score either when on the serve, the shuttlecock hits the proper side of their opponenet's service court, or at any point after the serve the shuttlecock hits any part of their opponent's court, or if when trying to return the shuttlecock, their opponent hits the shuttlecock out of bounds or hits the net. It becomes their opponent's turn to serve if they make a fault on a serve, the shuttle lands in their side of the court, or they hit the shuttle out of bounds or hit the net.
Keep on playing like that. A doubles and men's singles game goes to 15 points, ladies singles is to 11 points. If the game becomes tied at 14-all, or 11 for the ladies, the side which first scored the last pair of points can choose, if they want, to have the game go for an additional 2 points, to 17 or 13. The winner of a game serves first for the next game, and they players switch sides of the court. The winner of a match is the one who wins the best of three games.
History:
Games resembling Badminton was played over 2000 years ago in Greece, China, and India. The children's games of battledore and shuttlecock used shuttlecocks made of cork, covered in leather, with feathers attached with wax. Some of the more expensive shuttlecocks are still made this way, except with glue instead of wax. Duh:)
The name of the game came from House Badminton in Gloucestershire, which was the home of the Duke of Beaufort, where it was played in the late 19th Century after being brought from India by English officers who were stationed there.
The official rules were first written up soon after that, and the sport quickly spread throughout the world. The first international championship was held in 1949, and in 1992 the sport was added to the slate at the Summer Olympic Games.