Before each game, the umpire will ask each side which of the players will be serving or receiving. If there is no umpire, the opponents may ask, or more commonly, players just position themselves. At the start of a game, both server and receiver stand on their right side; their partners may stand anywhere on court as long as they don't obscure the opponents§9.5.
When the receiving side wins a rally, the players remain on the sides they were at the start of the last rally§11.3.2. The right to serve cycles through§11.4
- The original server (Alice)
- The partner of the original receiver (Bella)
- The partner of the original server (Alex)
- The original receiver (Bob)
If team B (Bob and Bella) win the rally, Bella will serve to Alex, from her left side.
If team A win the next rally after that, Alex will serve to Bella, from his left side.
If team B wins the next rally after that, Bob will serve to Alice, from his right side.
If team A wins the next rally after that, we have completed the cycle, and Alice will serve again to Bob, from her right side.
When the serving team wins a rally, the server switches sides and continues to serve§11.3.1. For instance, if team A wins the first serve, Alice will now serve again, from her left, to Bella. If team A wins the next rally, Alice serves from the right, to Bob. This continues on and on; Alice keeps serving until team B wins a point or the game is over.
If that sounds complicated, there is a very helpful rule to remember: If the score of the serving team is even, they serve from the right. If the score of the serving team is odd, they serve from the left. For example, if team B wins the first rally, the score is 1-0. Since 1 is odd, the team serves from the left, where Bella is standing at the moment, so she's serving.
If you prefer a more interactive demo, I've set up one here. Simply click
+1 to see what happens when one of the teams scores.
The original laws referenced throughout this answer, download them from the Badminton world federation. Note that this explanation applies to the current 3x21 and the experimental 5x11 rally-point systems - before 2006, the rules were different.