Weapon Mastery is a class feature shared between some martial classes that reflects their specialization with certain types of weapons. The number of mastery properties you can learn is specified in your class's version of the Weapon Mastery feature. Assassins, Blackguards, Fusiliers, Paladins, Rangers, and Tricksters can only access two mastery properties at a time. Barbarians and Dragoons start with two; Knights and Samurai start with three, and they all gain access to more mastery properties, as detailed in their Features table.
If you're wielding a weapon and have learned its mastery property, you'll be able to use that mastery property every turn when you make an attack with the weapon.
If a weapon has the Double weapon property, then its Mastery while it is being wielded as a double weapon is in brackets.
Different mastery properties have different triggers. For example, Cleave requires you to hit a creature with a melee attack roll while Graze triggers when you miss a creature with an attack roll.
Unlike some abilities, there's no limit to how many times mastery properties can be used per Short or Long Rest.
Cleave. If you hit a creature with a melee weapon attack, you can make a second attack against a creature within 2m that is also within your reach. When you hit with the second attack, you can roll your weapon's damage, but you don't add your ability modifier unless it's negative.
Graze. If you miss a creature with your weapon, you deal damage equal to the ability modifier you used to make the roll.
Nick. When you are about to make the extra attack for Two-Weapon Fighting using a weapon with this property (e.g. when using two Light weapons), you can make it as part of the Attack action instead of as a Bonus Action. You can make this extra attack only once per turn.
Push. You can launch a creature you hit (up to one Size larger than you) 4m away from you.
Sap. An enemy you hit has Disadvantage on their next attack roll before the start of your next turn.
Slow. When you hit a creature and deal damage, you can reduce its Speed by 4m until the start of your next turn. This speed reduction does not stack with itself.
Topple. When you hit a creature, you can force it to make a Strength saving throw (DC = 8 + the ability modifier used to make the attack + your Proficiency Bonus) or fall
Vex. When you hit a creature and deal damage, you gain Advantage on your next attack roll before the end of your next turn.