Mastery is a system for upgrading your powers. The Mastery system is unlocked by getting the Mastery achievement for clearing level 100 with 1 active power.
Each of your powers has Mastery equal to the highest level enemy you've defeated while you only have that power, and no other powers. Every 100 Mastery can improve certain effects of your power, usually by 10% of its base effects. Mastery and improvements based on Mastery stay on the power, even if you start a new run with time travel, and even if you lose and regain the power. Mastery on your powers doesn't improve your enemies' powers.
For example, defeating a level 100 enemy with Super Speed, and no other powers, increases your Super Speed's Mastery to 100, and Agility multiplier from 10 to 11. Defeating a level 200 enemy with only Super Speed increases its Mastery to 200, and multiplier to 12, and so on. Your Super Speed keeps these increases, even if you start a new run with time travel, and even if you lose and regain Super Speed.
In addition, total Mastery accumulated over all your powers can be spent on Rotation Powers.