The Lunar Forge can be found in Silent Moons Camp, at the top of the stairs and inside the small barrow.
Furthermore, several Lunar weapons can be found nearby the forge, including under the grindstone.
In order for the Lunar Forge to work, It must be under direct moonlight, thus the name Lunar Forge. A Lunar weapon can be disenchanted and the Silent Moons Enchant learned. This allows the unique enchantment to be applied to other weapons. Although it is stated in the game that the Dragonborn can craft lunar weapons at this forge when Tamriel's two moons align, this feature is not present in-game.