Coal

Coal lumps are items harvested directly from coal ore blocks, and as a drop from Wither Skeletons, making coal a renewable resource. They are the third most efficient fuel available for furnaces (after lava buckets 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).

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.

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.

Charcoal
Charcoal is an alternative to coal. Smelting wood in a furnace creates charcoal, which can be used for everything coal can (i.e., crafting items, powering minecarts, and smelting objects), except crafting coal blocks. 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, and they can be differentiated because they have different texture files (as of 13w19a).

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.

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.

Trivia

 * Stated by Notch in a livestream, charcoal is just a name class edit of coal, explaining why it functions like coal.
 * Coal and Charcoal have two different data values and thus do not stack with each other, even though they can be used interchangeably.
 * Charcoal was actually originally suggested by a user on Twitter as a writing implement.
 * In Minecraft Pocket Edition, when you touch Charcoal for the item name, it will say 'Coal'.
 * In the Pocket Edition, there are two separate crafting recipes for torches: one is for natural coal, and the other is for charcoal.
 * Coal can be found in chests in Mineshafts and Strongholds.
 * A Block of coal can smelt 80 items, which is more than the smelting time of its ingredients, 9 coal items.