To put the tennis scoring system as simply as possible, one must win:
We'll call the players A and B.
By winning a coin toss or a spin of the racquet, A gets to choose one of the following:
Let's say A chooses to serve. B then gets to choose an end of the court. A may serve from anywhere behind her baseline between the right singles sideline and the center mark. The serve must be struck before the ball bounces, and it must land in the service box diagonally opposite her. She gets two chances to get a serve in. If she misses both, she loses the point. If a serve that is otherwise good nicks the net on its way in, it is redone.
If A gets her serve in, B must return the ball, after exactly one bounce, into any part of A's singles court. A and B must then return the ball, after no more than one bounce, into one another's singles court until one of them misses.
A will serve from the left side of her baseline for the second point of the game, and she will continue to alternate right and left for the start of each point of the game.
Let's say A wins the first point. At the start of the next point, she must announce the score, her point total first: "15 - love." (Love = 0.)
B wins the next point: "15 all."
B wins the next point: "15 - 30."
A wins the next point: "30 all."
A wins the next point: "40 - 30."
If A wins the next point, she wins the game.
If B wins and it is "40 - 40" Players A and B play one more point to determine the winner of the game. Player B (The returner) gets to choose which side they want to receive the serve on - the Duece or Ad side of the court.