The following information applies to scoring in doubles matches.
Points are scored only on the serve. The receiving side cannot score a point. At the start of the game, the player on the right side serves. If a point is scored, the server moves to the left side and serves to the opposite court. Each time that a point is scored, the players on the serving side alternate from the right to left or left to right. That player continues to serve until the serve is lost by losing a rally. Players on the serving team do not alternate sides unless a point is scored. The receiving side never alternates sides.
When the receiving team wins a service turn, the player on the right serves first and continues to serve and alternates sides each time that a point is won. When the first server loses the serve, the serve goes to the second server on the side. When the second server loses the serve, the serve goes to the other team and the player on the right serves first. That pattern continues throughout the game.
Call the score in the order of server’s score, receiver’s score, then the server number. For example, if the score of the serving team is 3 and the score of the receiving team is 6, and the second server on the side is serving, the score would be announced as 3-6-2. Some players might say “first” or “second” for the server number, for example 3-6-second. Either way is correct. Note that the server number is for that service turn only. Whoever is on the right when the team gets the serve back is the first server for that service turn only. The next time that the team gets the serve back, it might be the other player that is on the right and is the first server for that service turn only. Beginning players often mistakenly assume that the player keeps the same server number throughout the game.
It is important to call the score including the server number prior to each serve. That helps the players to keep the server and the score straight. When you announce, as part of the score, that you are the first server, everyone on the court knows that when you lose the serve, the serve goes to your partner. When you announce that you are the second server, everyone knows that when you lose the serve, the serve goes to the other team. To minimize the advantage of being the first team to serve in the game, only one player gets a service turn on the first service turn of the game. To help everyone remember that the serve goes to the other side when that player loses the serve, it is customary to say that the player is the last server or second server. Therefore, at the start of the game, it is customary to say that the score is 0-0-2. The “2” means that the serve goes to the other side when the serve is lost.
At the start of each game, make a mental note of the player that served first for each side. If the rotation is done correctly, a team’s score will always be even when that player is on the right and odd when that player is on the left. As you call the score, use the player position as a double check on whether you have the correct score.
Singles scoring is very similar except that there is no second server. The serve is always done from the right side when the server’s score is even and from the left side when the server’s score is odd. Note that it is the server’s score that matters, not the combined score of server and receiver.