Hopper

Hoppers are blocks that were added in Update 0.14.0. They are used to catch items, or to transfer items into and out of Containers such as Chests.

Crafting
5 Iron Ingots + 1 Chest => 1 Hopper

Uses
Hoppers are mainly used as a method of transferring items from one container to another, and to catch any item entities that fall on it.

Interaction
When connecting hoppers the player needs to sneak/crouch before placing the hopper on the face of the container it should connect to. If not crouching the player will access the hopper's inventory.

Hoppers interact with different containers in different ways:

Hoppers
Hoppers can be connected to hoppers enabling item transportation. This allows for transportation sideways and downwards.

Droppers
When connecting a Hopper to a Dropper facing upwards, one can with some redstone circuitry transport items upwards.

Mobs
The player could have a mob follow them to a Hopper, then kill them and have the Hopper pick up the mob drops.

Chests
Any Hopper attached to a Chest will push an item from its own inventory into the Chest. If a Chest is above the Hopper, it will pull the item from the Chest into its own inventory.

Furnaces
A Hopper attached above a Furnace will fill the Furnace's ingredients slot. A Hopper next to a Furnace will only fill the fuel slot. A Hopper below a Furnace will only collect items from the output slot.

Brewing Stands
A Hopper above a Brewing Stand will only fill the top ingredients slot with Potion ingredients. A Hopper to the side of a brewing stand will only push Water Bottles and Potions into the bottom three Bottle slots, and Hoppers below the stand will collect any resultant Potions.

Dispensers
A Hopper above a Dispenser will fill it. Hoppers below Dispensers will remove items from them.

Minecart With Chest
When above a Hopper (Rails can be placed on Hoppers), the Hopper will take items from the Chest.

Trivia

 * Hoppers can be used for automation, as well as in transporting of items over long distances.
 * Hoppers will not function if a Redstone power source (like a Redstone Torch) is placed adjacent to them.
 * Hoppers can pick up 64 items at a time; however, they can only transfer it to another inventory 1 at a time.
 * Hoppers only have five inventory slots.
 * The containers where the hopper transfer the items depends on which blocks of the bottom half of the hopper connects.
 * Hoppers can transfer non-smeltable items/non-fuel items into the furnace.
 * A Hopper is a Container.
 * Since Update 0.15.0, players can stand in Hoppers and Cauldrons.