Stars on ping pong balls represents the quality of the ping pong ball. They go from a scale of 1 star to 3 stars where 1 is the lowest and 3 is the highest representing that the ball was made from high quality plastic and bounces better.
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The higher amount of stars the better quality and bounce the ball has. One to three stars is for people who just want to have fun, and three to five stars on a ball is for the serious player.
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The above answer is partly true. I've been in organized table tennis in the US for over four decades. So I'd like to improve on the above answer by saying 0 to 3 stars is the most I've seen balls rated, with 3-stars being the best for a recreational ball.
As for balls used for competition or sanctioned play, not only will they have 3-stars but they will also have the ITTF (International Table Tennis Federation) letters on them too. The ITTF is the world governing body for international tournaments like the one that is taking place right now in Russia, the WORLD TEAM CHAMPIONSHIPS. you can go to www.ittf.com for more info.