The badminton scoring system discussed on this page is based on the 21 points or known to many as the new rules.
Whether it's badminton singles or doubles, whichever team reaches 21 points first will be the winner of the game.
You get a point by winning a rally.
You win a rally if the shuttle drops within the INSIDE boundaries of your opponent's side of the court.
Once you're awarded 1 point, you're deemed the winner of a badminton rally.
The winner of a rally will be awarded 1 point, and then starts the next rally by making a badminton service.
You should also know which area on the court you should be standing when making a service. It's very simple.
The player making a service serves diagonally to the opponent's side.
If both players deuce at 20-20, counting of points will continue even after 21 points. A player will ONLY be announced as the winner if he/she leads by 2 points (eg. 22-20, 23-21, 24-22...)

Note: See badminton rules for detailed information on the "in" or "out" boundaries for a badminton serve.
The player with an:
A simple illustration is provided in the picture below.

The serving position for badminton doubles is pretty much the same with badminton singles. The only difference is that one side will have 2 players.

I'll just provide a simple example to explain it clearly.

Still don't get it?
Don't worry! It's pretty simple. Here are examples of 2 possible scenarios...
One team will have 2 players (player A and Player B). The score is 0-0 and the team will start the game by making the first badminton service.
If the team WINS the rally, the point will be 1-0 and "player A" will have to move to the left hand side of the court. "Player B" moves to the right hand side (i.e. both of them switch places).
Now, "Player A" will make the service from the left side.
Then during the next rally, "player A's" team WINS THE POINT BACK and gets 1-1. This time, "player B" serves** instead of "player A" (They DO NOT need to switch areas).
**Since 1 is an odd number, the player standing at the left hand side of the court will make the service.