A high score is the highest score there is for a game. It's what's currently the best score and what other players must beat.
Birth of the High Score
Back when arcades were popular during the Golden Age of Arcade Games, many games were based on getting a high score, a concept popularized by Space Invaders. Arcade games had little to no story, only gameplay and for that gameplay to have a goal, the ranking system was invented. This was meant to keep track of a players score. The ranking system usually showed off only the ten best player's scores and whoever was number one had the high score. In certain games, trying to get the high score was part of the instructions. This made arcade gaming much more competitive as players were fighting for bragging rights. Adding to this was the fact that you could add your initials to your score to boost your personal ego. Those who cared did this right, the others usually just put "AAA" or "ASS". Some arcades even wrote the names of the people who held certain high scores on a chalk board behind the counter so everyone could see.
High Scores Today
Despite the "death" of arcades, high scores still exist and are as popular as ever. Thanks to online gaming, players can share there high scores on online leaderboards and compare there score to the rest of the world. Some recent arcade games like After Burner: Climax still encourage you to go for that high score and iPhone gaming is basically based on getting one. Doodle Jump is a great example of the fact that high scores still mean something as it is one of the most downloaded iPhone games to date.
Microsoft's Game Room makes special use of the high score mechanic through its Challenge system, allowing players to go head-to-head to achieve the highest score in the game or specific situation of their choosing.