Armor

The Dungeon Master’s Guide tells us about Adamantine Armor on page 150.

Adamantine Armor

Armor (medium or heavy, but not hide), uncommon

This suit of armor is reinforced with adamantine, one of the hardest substances in existence. While you’re wearing it, any critical hit against you becomes a normal hit.

Weapons

There is more information about Adamantine Weapons in *Xanathar’s Guide to Everything* on page 78:

Adamantine is an ultrahard metal found in meteorites and extraordinary mineral veins. In addition to being used to craft adamantine armor, the metal is also used for weapons. Melee weapons and ammunition made of or coated with adamantine are unusually effective when used to break objects. Whenever an adamantine weapon or piece of ammunition hits an object, the hit is a critical hit.

The adamantine version of a melee weapon or of ten pieces of ammunition costs 500 gp more than the normal version, whether the weapon or ammunition is made of the metal or coated with it.

So, it affects hardness, cost, and damage. Adamantine armor turns critical hits against the wearer into regular hits; any hit from an adamantine weapon against an object does extra damage because it’s a critical hit.

There are also a handful of monsters (mostly object like ones) where their non-magical damage resistance is negated by adamantine, like the gargoyle:

Damage Resistances: Bludgeoning, Piercing, and Slashing from Nonmagical Attacks that aren’t Adamantine