The player score is a match specific statistic that begins with a value of 0 upon first spawning in and can be increased and decreased by various methods depending on the game mode.
In Survival Mode, you gain points by destroying asteroids with your own lasers, getting kills and picking up gems. Revenge killing someone will give you extra points. You lose score by either losing gems or dying.
In Team Mode, the methods to gain and lose score are the same as in survival mode, except you can also donate gems to the station to increase your score. The amount of points you gain by donating gems to the station is equal to the amount of gems donated. Picking up gems given by a teammate using "V" will not give you any points. Killing aliens at the star also doesn't add to your score. As of the 10-25-2019 update healing teammates now gives you points depending on how much you heal at a rate of every 10 shields you get a point.
In Pro Deathmatch, the only way to gain score is to get kills. You get 2 points for killing your target and 1 point for killing other people. There is no method of losing score in Pro Deathmatch.
In Invasion, you gain score by killing aliens and picking up gems. You and all teammates will also gain score after completing a wave. Destroying asteroids in this mode will not give you any score. The only method to lose score in invasion is to lose gems; dying will not cause you to lose score.