The absolute basics
The aim of badminton is to hit the shuttle with your racket so that it passes over the net and lands inside your opponent’s half of the court. Whenever you do this, you have won a rally; win enough rallies, and you win the match.
Your opponent has the same goal. He will try to reach the shuttle and send it back into your half of the court. You can also win rallies from your opponent’s mistakes: if he hits the shuttle into or under the net, or out of court, then you win the rally.
If you think your opponent’s shot is going to land out, then you should let it fall to the floor. If you hit the shuttle instead, then the rally continues.
Once the shuttle touches the ground, the rally is over. In this respect, badminton is not like tennis or squash, where the ball can bounce.
You must hit the shuttle once only before it goes over the net (even in doubles). In this respect, badminton is not like volleyball, where multiple players can touch the ball before sending it back over the net.
Serving is how you start the rally: someone has to hit the shuttle first! To prevent the server gaining an overwhelming advantage, there are special restrictions placed on serving that don’t apply during the rest of the rally.
The receiver is the person who hits the second shot in the rally. In doubles, the receiver’s partner is not allowed to hit this shot.