Elytra is a special kind of Armor, introduced in the 15w41b snapshot, which takes up the chestplate slot.
Elytra are a pair of wings that allow you to glide from high places. They are considered an end-game Item. Elytra will allow a 10-1 glide ratio, which means that you can go 10 blocks forward for every block down at maximum efficiency. To use it, you must first equip it, then jump and hold while falling.
Elytra can be repaired with leather on an anvil. They can also be repaired by placing two damaged elytra side by side on a crafting table. Their Durability will be added together, and the two elytra will be merged into one).