RuneScore is a points system used to measure completed achievements. It was added on 29 August 2017 and can be viewed by clicking the Hero icon in the game interface; it is located on the upper left tab. Most achievements have an assigned value (between 5 and 75) and upon completing the achievement its value is added to the player's RuneScore.
The maximum possible RuneScore is currently 21,570, coming from 1,947 achievements, as of 22 August 2018. The game unfortunately shows different amounts for both the total of achievements as well as the maximum RuneScore. Jagex is aware of this issue.
Some achievements that are heavily luck-dependent, such as skill pets, boss pets or boss collection log achievements, do not contribute to a player's RuneScore and have an assigned value of 0.