Leather is the tanned hide of an animal. It can be used to make bags, quivers, backpacks, clothing, shields, armor and water skins at a leather works. It can also be used to decorate there. However, leather crafts can only be made at a craftsdwarf's workshop.
Leather armor will be worn by civilians with the hunting labor enabled, or by military dwarves with at least leather armor enabled. Dwarves instructed to wear chain and plate armor will still wear leather if chain and plate armor are unavailable. Leather is significantly lighter than other forms of armor, and thus is less likely to cause encumbrance penalties.
Making leather requires a tanner's shop, a raw hide or skin from an animal, and a dwarf with the 'tanning' skill turned on. Rotten hides or skin cannot be tanned. The human and dwarven caravans usually sell various leather hides too, in case few animals inhabit the map. One unrotten hide yields one "unit" of leather - regardless of the animal being a kitten or an elephant. Dwarven tanners have a skill unknown to mere humans in that they can make the softest skin of the tiniest animal equally hardy as that of the most heavily protected live creature. Whether they make flimsy skin into a rock-like substance, or tough hide silky and supple, remains a mystery.
Leather of rare or dangerous animals can be quite valuable. The "leather" from giant cave spiders and giant desert scorpions is called "chitin", but it is otherwise processed and treated as leather in all ways.