Coal

Coal lumps are items harvested from coal ore blocks. They are the fourth most efficient fuel available for furnaces (after Lava Buckets, Coal Blocks, and Blaze Rods), being able to smelt eight units per lump of coal. Coal lumps and charcoal are also the only fuels accepted by powered minecarts (providing approximately four minutes of transit each).

Charcoal is an alternative to coal. Smelting wood in a furnace creates charcoal, which can be used for most of the same things coal can, except for crafting blocks of coal. Charcoal makes it easier for players to survive their first night as it allows them to craft torches much more easily. Charcoal does not stack with coal.

Aside from coal having the ability to craft blocks, coal and charcoal are functionally identical. They otherwise differ only in their data IDs, images, and in that they can't be stacked together. Coal and charcoal can each smelt with equal efficiency, and craft the same number of torches and fire charges.

Obtaining
The main difference between coal and charcoal is in the methods of obtaining.

From ore
The most common method of obtaining regular coal is by breaking a block of coal ore, which drops one coal item, or more if broken with a pickaxe with the Fortune enchantment.

Coal ore can be found at any elevation, wherever there is stone. It is the most abundant ore, and can be found in veins of 5 or more, up to 32. Coal has a 2/3 chance of dropping experience when mined.

Smelting
Smelting coal ore is wasteful and usually pointless: The ore could be placed and mined to get at least as much coal, with no fuel needed. (Smelting always yields a single unit, but mining with a Fortune-enchanted pickaxe can get more.) Coal ore as an item can only be obtained by using a Silk Touch pickaxe.

The second recipe makes charcoal a renewable resource. If the player is on a superflat map that doesn't contain ores, wood blocks from NPC Villages can be smelted into charcoal. This allows for the creation of torches, which are otherwise in short supply.

From Wither Skeletons
A Wither Skeleton may drop a coal lump upon death. This makes regular coal a renewable resource.

As a crafting ingredient
The only purpose for which coal could be considered required is the creation of torches, which are essential for prolonged underground exploration — although there are other permanent light sources, none come close in terms of convenience and availability. Carrying a full stack of coal and 8 Wood allows you to craft up to 256 torches. Perhaps the next-best alternative to coal-based lighting would be Netherrack, which, if manually ignited, burns forever. They can also be used to make Blocks of Coal, a good fuel to smelt.

As a fuel
When used in a furnace as a fuel, a coal lump or piece of charcoal lasts 80 seconds (8 operations).

Coal lumps and charcoal are also the only fuels accepted by powered minecarts (providing approximately four minutes of transit each).

Efficiency
Wood and wood products can be used for smelting directly. A more efficient (but also more time-consuming) approach is to smelt wood into charcoal, which can be used for smelting. Wood and wood product smelting yields are shown in the table below.

It is fairly efficient to turn one wood block into wood planks, and then use those planks to smelt more wood into charcoal. This process can smelt 6.85 items per wood block.

It is more efficient to turn wood blocks into charcoal, and use those to smelt more wood into charcoal. This can smelt 7 items per wood block.

It is most efficient to use two saplings (which are a free by-product of chopping wood) to smelt a wood block into charcoal, and then use the charcoal for smelting. This process does not waste any wood, and therefore can smelt 8 items per wood block.

Video
Note: The video does not show a crafting recipe for Blocks of Coal. This was before Blocks of Coal were added in 1.6.

Note: The video states that charcoal looks the same as coal. This was before 1.6, where charcoal's texture was changed.