The Green Gem is an item used to craft several Ancient items. It can be used to craft the Construction Amulet, which raises Sanity as well as halves the resource cost of crafting items, and the Deconstruction Staff, which can be used to destroy structures/items and receive all of the resources used in their construction (barring Gems).
The Green Gem's element appears to be labour; its opposite being the Orange Gem.
The Green Gem uses the Red Gem's texture, only flipped and coloured green.
In the Reign of Giants DLC, because Green Gems are (barely) renewable, the items it can be crafted into can be utilized to make nearly any other item in the game renewable, by crafting something with the amulet at half-price and then deconstructing it with the staff for a full return.