There is no way to make a Saddle in Minecraft. The player will only be able to get a saddle by finding them in chests. These chests can be found inside dungeons, abandoned mines, nether fortresses, jungle temples, desert, blacksmith chests inside NPC villages, trading with villagers, or even by Fishing.
Now that you have learned how to get a Saddle, it can be placed on a pig, mule, donkey, or horse and allow the player to ride them by right clicking the saddle itself. A Carrot on a Stick is required to control the movement of a pig. Killing any of the animals will drop the saddle but for a horse you are able to shift right click to open an inventory box to dismount the saddle from the horse.