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
Saddles may be used to ride horses and pigs.

a saddle on a pig will equip the saddle, allowing the player to ride the pig, though controlling the pig requires usage of a carrot on a stick. The saddle can only be retrieved by killing the pig.

Horses can be ridden without them wearing a saddle, but a saddle is required to control them. Horses can be equipped placing it in their inventory, which can be opened by pressing while holding a saddle or. The saddle can be unequipped by removing it from the horse's inventory.

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Note: video is out of date - saddles can now also be used with horses.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