Leather Armor is made out of the skin of a animal/human (usually animal though). It offers minor protection but is cheaper and lighter than metal armors. Unlike leather clothing, leather armor is very stiff, allowing it to provide greater protection to the wearer. In many games leather armor is worn by sneaking classes like a rogue or an assassin because It is lighter and doesn't make as much noise as metal.
Some games also restrict the abilities of certain characters depending on the armor they wear - in any Dungeons and Dragons based game, a dual-classed thief/mage may be able to physically wear armor, but doing so will leave them incapable of the range of motion necessary to cast magic spells.
Most games offer studded leather as a variant on plain leather armor, with the inclusion of metal rivets for additional protection (and additional weight).