Some Mastery Properties require your target to make a saving throw to resist the effects. Unless otherwise stated, the saving throw DC is calculated as follows: 

Mastery save DC = 8 + your proficiency bonus + your Strength or Dexterity modifier (your choice) 

Each weapon and shield has 4 "Mastery Properties" associated with it. The underlined property is known as the "Signature Property" while the other 3 are "Additional Properties". When you acquire a feature for mastering a weapon it will specify which properties you can learn.

There is one Mastery Property that is unique among the others known as "Specialist". This property can be taken when specified on the feature giving you masteries. It is also the only mastery that can be taken up to twice on each of your weapons.

There are multiple ways to obtain masteries. All martial classes now utilize masteries as a core mechanic, as well as some half-casters and "third martial" subclasses such as Bladesinging Wizard. There are also new mastery focused Feats.