There are two species of Elephants: Asian Elephants, African Elephants; and two of Mammoths: Woolly Mammoth and Songhua River Mammoth. The Mammoths only spawn in the Tundra biomes and the elephants spawn in the desert, plains or forest.
Elephants and Mammoths can be tamed by feeding the calves ten Sugar Lumps or five Cakes, but this will only work when they are Calves. Once you have fed the baby Elephant/Mammoth the food, the naming screen will appear. If you wish to rename your Elephant/Mammoth, you can right-click on it with a Book, a Medallion or a Name Tag.
Tamed Elephants can be led around on a Lead, and healed by feeding them Bread, Hay Stacks,Baked Potatoes Or Regular Potatoes.
Be careful with how you keep your Elephant or Mammoth. Hostile Mobs will attack a tamed Elephant/Mammoth, although even the calves will fight back. You should probably have high walls and a roof to keep out scorpions.
An Elephant Harness can be placed on tamed, adult Elephants/Mammoths in order to make them rideable and to be able to add extra items to the Elephant/Mammoth. It is created using Hide, Wool and Iron, but is only accessible for 1 player.
If a player 'sneaks' near their Elephant/Mammoth ('shift key' by default), it will sit for a short time, allowing the player to sit in the Harness by right-clicking on it. To dismount an Elephant/Mammoth, press the shift key once more. Note, you cannot add any of these to calves.
An Elephant Howdah is a kind of throne that can only be applied to a tamed, adult Asian Elephant. It is a purely decorative item. A Howdah can only be applied once an Elephant Harness and an Elephant Garment have been applied.
Chest Sets can be applied to tamed adult Elephants and Mammoths in order to give them their own inventory, much like Horses, Mules, Wyverns and Ostriches. Two Chest Sets can be applied to each Elephant.
Chest Sets can only be applied once an Elephant Harness has been given to the Elephant/Mammoth. Once applied, a Key will appear in the player's inventory. Woolly Mammoths can carry two extra regular chests.
Tusks are reinforcement for the Elephant/Mammoth. When wearing Tusks, Elephants and Mammoths are able to break blocks that are in their path. There are three different kinds of Tusks: The Wooden Tusks, the Iron Tusks, and the Diamond Tusks. Tusks are applied by right-clicking on the tamed adult Elephant/Mammoth with the Tusks in hand, and can be taken away by right-clicking on the Elephant/Mammoth with a Pickaxe.
Mammoth Platforms can be placed on Songhua River Mammoths in order to allow them to carry a second player. In order to have a passenger, you need to first make the Mammoth sit by pressing the sneak key ('shift' by default), then the second player 'sneaks' close to the Mammoth and right-clicks on it to climb aboard.