HUD
The HUD (Heads Up Interface) is a small menu in the gameplay screen. It is located near the bottom of the screen. It contains the instant inventory, health bar, hunger bar, air bar (only shown when underwater), and armor bar. The instant inventory can be navigable with the 1-9 number keys for instant selection of tools.
Crafting
When you aren't using a crafting table, there is a 2x2 grid.
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When you are using a crafting table, there is a 3x3 grid.
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Smelting
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When you shift-click on the furnace (pictured above), you will see an arrow, a rectangle, and three slots. The arrow indicates progress. The rectangle displays how much fuel is left. In the upper box, insert the resource you wish to smelt. In the lower box, insert the resource you wish to smelt with (such as coal, lava, sticks, etc.) The box to the right is where you will receive your smelted resource.
Chest
When using a chest you will see two large boxes. The lower box is your inventory, and the larger box contains 27 additional slots with which your chest is comprised of. More at page: chest.