Saddle

Saddles are items that can be placed on ridable mobs.

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, 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 steed's inventory.

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 causes the mob to wear the saddle, and it will be taken out of your inventory, even in creative mode.