Saddle

Saddles are items that can be placed on ridable mobs.

Natural generation
Saddles can be found in chests inside dungeons, abandoned mine shafts, Nether fortresses, desert temples, jungle temples and in blacksmith chests in villages.

Fishing
Saddles can be obtained as a "treasure" item from fishing.

Trading
Villager leatherworkers will sell saddles for 8-10 emeralds as a third tier trade.

Usage
To put a saddle on a pig, press use button on a pig while holding a saddle in the hand and the pig is equipped with a saddle. The only way to retrieve the saddle is to kill the pig.

To put a saddle on a tamed horse, press use button on a horse while sneaking. The horse inventory will be opened, and the player can put the saddle into the saddle inventory slot of the horse. The player can also unequip a horse by removing the saddle from the saddle inventory slot of the horse. Saddles cannot be equipped on an untamed horse.

In order for the player to ride a pig or control a horse's movement, they must be equipped with a saddle. Pressing use button on a pig equipped with saddle, and the player will ride the pig. However, to control the pig's movement, a carrot on a stick is also required. Pressing use button on a horse equipped with saddle, and the player will ride the horse and be able to make it move in all directions and jump.

Video
Note: video is out of date - saddles can now also be used with horses.

Trivia

 * One of the splashes which shows up on the title screen reads "Ride the pig!".
 * If the pig the player is riding moves under low solid blocks, it is possible they can phase the player through these blocks, damaging the player through suffocation. This is similar to how the spider in a Spider Jockey will usually suffocate the skeleton if they spawn in caves.
 * When riding a pig or horse, block mining time is reduced considerably.
 * It is not possible in SMP to ride a pig and have another player control the pig by walking in front of it while holding a carrot.
 * If you ride a pig off of a cliff, you can dismount from the pig before it hits the ground and take little to no fall damage, but the pig will still die. This can be very useful when trying to get the "When Pigs Fly" achievement.
 * If lightning strikes a pig while you are riding it, you will take damage, you will be dismounted and the pig will turn into a zombie pigman.
 * The player's body seems to be always facing the way the pig does, ignoring the direction player is looking. Switching to third person perspective allows to see only player's head rotating. This has not always been this way, it used to work just like minecarts.
 * A saddle is considered "armor" by the game, but the value of protection to the pig is equal to null, making no additional protection to the pig. A saddle can still be seen if the pig is invisible.
 * Punching a pig while holding a saddle causes the pig to wear the saddle, and it will be taken out of your inventory, even in creative mode.
 * Saddles were added in Pocket Edition in the 0.7.0 update, but were only accessible by inventory editing and do not function. As of 0.9.0, they can be found in dungeons, but still do not function.
 * However there is riding code in Pocket Edition.