Saddle

Saddles are items that can be placed on pigs and horses, which allow the player to ride them by right-clicking with the saddle. Pigs require a carrot on a stick for controlled movement, while horses do not. Saddles cannot be placed on any other mobs. The player can dismount the pig or horse by pressing left-shift, and the saddle would remain on the animal indefinitely. Killing the animal that has the saddle on would drop the saddle.

Saddles are only found in chests inside dungeons, Nether Fortresses (Minecraft 1.6) and Desert Temples (Minecraft 1.6), or by trading with a villager. Prior to horses being added, Dinnerbone stated he will not be including a crafting recipe for saddles.

Pig carting


The player can continuously ride a pig in a minecart without stopping. Pig carting is when a player puts a saddle on a pig and pushes the pig into a minecart. The pig can then be ridden while it is in the cart, allowing a self-propelling minecart ride. The player on the minecart with the saddled pig travels significantly faster than a regular minecart with a mob or player on it. The player can actually overlap the powered minecart and rebound off of it to go in the opposite direction before rebounding from that direction and going back to the original direction. This will happen continuously if the tracks are a closed loop. If a saddled pig rides a minecart and the player mount the pig, the player can control it, moving at a slow speed, with the left and right strafe keys. The pig will continue to move in much the same fashion if it is moved off the rails, thus making a "car". However, this is rather ineffective, as the car, although it doesn't deplete your hunger, will only move pig speed. You can steer with a Carrot on a Stick. This is very effective as there is no need to waste gold to build powered rails when you can just use a pig and a saddle.

Horse Saddle
Horse saddles were items that were added along with horses that worked just like regular saddles, but for horses instead of pigs. Taming a horse was required before putting a horse saddle on it.

Horse saddle items were later removed from the game, and the same saddles (found in dungeons) are now used for both pigs and horses.

Horse saddles were crafted as follows:

Issues
You can't make horse saddles any more. They only work in the first few 1.6 snapshots. You will need to find a regular saddle.

Trivia

 * One of the splashes which shows up on the title screen reads "Ride the pig!".
 * By throwing chicken eggs or snowballs at the pig while riding it, or simply punching it, it used to be possible to ride a pig high into the air. However, this is tricky to pull off because right clicking on the pig to throw something may dismount the pig. This has since been patched.
 * It would probably be easiest to pull this off by throwing these items upwards, however it is likely you will miss a few.
 * 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 on a saddle, blocks are destroyed slower than on foot, at roughly the same speed as when underwater.
 * Since passive mobs no longer despawn as of beta 1.8, placing a saddle on a pig no longer wastes the saddle when you walk away, as long as the pig can not be harmed.
 * It is not possible to throw Ender Pearls while riding a pig, but if players look down at the pig and throw a Pearl they will be dismounted from the pig and take the Ender Pearl's damage.
 * 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, and the pig will be demounted and turned into a zombie pigman.
 * 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, despite providing no 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.
 * It is possible to obtain saddles with the "Generate Structures" option turned off, by curing zombie villagers and trading with them. (It is recommended to breed them when you have two villagers. This method also works for obtaining Bottles o' Enchanting when "Generate Structures" is turned off.)

Gallery
*Notice that the Pig model changed in Beta 1.8, their noses now protrude from their faces.