Score
In a game the score refers to the amount of points achieved by a player or team.
In almost all games a high score of many points is what is needed to win the game. However, there are a few notable exceptions.
Sport
In many popular sports points are most commonly achieved through obtaining "goals" or "scores." For example, in soccer, hockey, and basketball goals are achieved by putting the ball in the opposing team's net. Other team sports like baseball and cricket have more compliated scoring procedures.
Individual-based sports like golf and tennis have points-based scoring as well.
Most sports have time limits, which means point-based victories are usually the result of obtaining more points than one's opponent, and not simply achieving a set "magic number" of points.
Video Games
In video games points are usually an optional, side component of gaming. Players may achieve points through normal gameplay, but their score will often not have an immediate relevance to the game itself. Instead, getting a "high score" and then trying to beat that score in subsequent play becomes a side-challenge to offer replay value.
A screenshot from the Nintendo game
Mega man. The score can be seen at the top of the screen. No subsequent
games in the Megaman series kept track of the player's score.
Some have argued that points in video games are somewhat of an anachronism in modern gaming. During the era of arcade games, when games could not be "won" or "completed," but were instead endless cycles of continious gameplay, points had a much greater relevance. Today, however, many gamers no longer care about beating high scores since such victories offer little reward other than self-satisfaction. Many modern games no longer even keep track of score, and many that do no longer feature an option to save record high scores.
Sometimes the score of a game can have relevance to gameplay. In fighting games, for example, scoring a very high number of points could result in unlockable players or modes. In arcade games or certain platformers, high scores could result in an extra life.