What weapon properties do
Properties define how a weapon is used: which ability it attacks with, how many hands it needs and how far it reaches. Every property in the equipment tables links here.
Ammunition
You can use a weapon that has the Ammunition property to make a ranged attack only if you have ammunition to fire from it. The type of ammunition required is specified with the weapon's range. Each attack expends one piece of ammunition. Drawing the ammunition is part of the attack (you need a free hand to load a one-handed weapon). After a fight, you can spend 1 minute to recover half the ammunition (round down) you used in the fight. The rest is lost.
Weapons with this property: Blowgun, Hand Crossbow, Heavy Crossbow, Light Crossbow, Longbow, Musket, Pistol, Shortbow, Sling
Finesse
When making an attack with a Finesse weapon, use your choice of your Strength or Dexterity modifier for the attack and damage rolls. You must use the same modifier for both rolls.
Weapons with this property: Dagger, Dart, Rapier, Scimitar, Shortsword, Whip
Heavy
You have Disadvantage on attack rolls with a Heavy weapon if it's a Melee weapon and your Strength score isn't at least 13 or if it's a Ranged weapon and your Dexterity score isn't at least 13.
Weapons with this property: Glaive, Greataxe, Greatsword, Halberd, Heavy Crossbow, Lance, Longbow, Maul, Pike
Light
When you take the Attack action on your turn and attack with a Light weapon, you can make one extra attack as a Bonus Action later on the same turn. That extra attack must be made with a different Light weapon, and you don't add your ability modifier to the extra attack's damage unless that modifier is negative. For example, you can attack with a Shortsword in one hand and a Dagger in the other using the Attack action and a Bonus Action, but you don't add your Strength or Dexterity modifier to the damage roll of the Bonus Action unless that modifier is negative.
Weapons with this property: Club, Dagger, Hand Crossbow, Handaxe, Light Hammer, Scimitar, Shortsword, Sickle
Loading
You can fire only one piece of ammunition from a Loading weapon when you use an action, a Bonus Action, or a Reaction to fire it, regardless of the number of attacks you can normally make.
Weapons with this property: Blowgun, Hand Crossbow, Heavy Crossbow, Light Crossbow, Musket, Pistol
Reach
A Reach weapon adds 5 feet to your reach when you attack with it, as well as when determining your reach for Opportunity Attacks with it.
Weapons with this property: Glaive, Halberd, Lance, Pike, Whip
Thrown
If a weapon has the Thrown property, you can throw the weapon to make a ranged attack, and you can draw that weapon as part of the attack. If the weapon is a Melee weapon, use the same ability modifier for the attack and damage rolls that you use for a melee attack with that weapon.
Weapons with this property: Dagger, Dart, Handaxe, Javelin, Light Hammer, Spear, Trident
Two-Handed
A Two-Handed weapon requires two hands when you attack with it.
Weapons with this property: Glaive, Greataxe, Greatclub, Greatsword, Halberd, Heavy Crossbow, Lance, Light Crossbow, Longbow, Maul, Musket, Pike, Shortbow
Versatile
A Versatile weapon can be used with one or two hands. A damage value in parentheses appears with the property. The weapon deals that damage when used with two hands to make a melee attack.
Weapons with this property: Battleaxe, Longsword, Quarterstaff, Spear, Trident, War Pick, Warhammer