How do appeals work, and when do I call the referee?

Under Law 17 of the BWF Laws of Badminton:

• An official's decision is final on points of fact they're responsible for (e.g. a line judge on their line, the service judge on service faults), except that you overrule a line judge who has clearly made a wrong call, and an Instant Review System decides challenges where one is used.
• You give a decision on any appeal about a point of dispute, as long as it's made before the next serve is delivered.
• You refer to the referee any unsatisfied appeal on a question of law only. It must be made before the next serve or, at the end of a match, before the appealing side has left the court.
• You also record and report to the referee everything relating to Law 16 (continuous play, misconduct and penalties).

The referee is in overall charge of the tournament; you're in charge of the match, the court and its immediate surrounds, and you report to the referee.