Saddle

Saddles are items that can be placed on ridable mobs with the exception of skeleton horses, zombie horses, and llamas, which controls itself and can be designed with carpets.

Fishing
Saddles can be obtained as a "treasure" item from fishing with a base chance [without fishing rod enhancements] of 0.8%.

Trading
Leatherworker villagers will sell saddles for 8-10 emeralds as a third tier trade.

Illager Beasts
Illager Beasts may drop one saddle when killed, and this drop chance can be increased with the looting enchantment.

Usage
Saddles may be used to ride horses, donkeys, mules, 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.

Tamed horses, donkeys, and mules can be ridden without them wearing a saddle, but a saddle is required to control them. Once tamed, they can be equipped with a saddle by 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 specific inventory slot.

Trivia

 * 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 or horse while holding a saddle caused the mob to wear the saddle, and it will be taken out of your inventory, even in Creative mode.
 * The "When Pigs Fly" achievement requires "Cow Tipper" even though you do not need leather to obtain saddles.