Coal

Coal lumps are harvested directly from coal ore blocks, which must be destroyed using any level of pickaxe. After this, they cannot be reverted back into block form.

They are the most efficient fuel available for furnaces, being able to smelt eight units of ore 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 allows you to craft up to 256 torches (assuming you have at least eight wood with you), and odds are you will be harvesting more coal as you dig). Perhaps the best alternative to coal-based lighting would be Netherrack, which, if manually ignited, will burn forever.

Coal ore can be found at any elevation, wherever there is stone. It is the most abundant resource, and can be found in veins of 5 or more (sometimes up to 70).

As a Crafting Ingredient
The items for a Fire Charge can be put anywhere on the crafting grid.

Charcoal
Charcoal is an alternative to coal. Burning wood in a furnace creates charcoal, which can be used for everything coal can, including making torches, powering minecarts, and smelting objects. This will undoubtedly make it easier to survive your first night as it allows the player to make torches much more easily. Charcoal does not stack with coal, though the two look identical and cannot be differentiated with a texture pack (They share the same texture.)

If you are using a superflat map, where there are no 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
It is more efficient (but also more time-consuming) to turn wood into charcoal than to use wood for smelting. A wood block smelts six items if broken into planks: each wood block makes four wooden planks, and each plank can smelt 1.5 items. However, one wood block can be smelted into one charcoal block, and each piece of charcoal smelts eight items, just like coal.

It is more efficient to take seven wood blocks, turn one into planks, and use those four planks to smelt the other six unbroken wood blocks into charcoal. These six charcoal can smelt 48 items. Seven wood blocks would only smelt 42 items, so this process can smelt 14% more items.

Moreover, if you begin with a single coal or charcoal, then you can smelt 8 wood blocks and make 8 charcoal, replace your starting charcoal and keep the other 7, making for a total of 56 smelts from those 8 wood blocks, though this takes even more time.

Other methods of creating Charcoal without coal:

Craft one wood block -> planks -> one set of 4 sticks. Use 2 of the sticks to smelt your remaining raw wood into charcoal.

Alternatively, the trees you cut the wood from will likely have dropped saplings. Two saplings, like two sticks, can be burned to produce 1 charcoal, without using up any of your precious wood.

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.

Data values
Kohle (Item) Charbon (objet) Szén (tárgy) Steenkool (voorwerp) Węgiel (przedmiot) pt-br:Coal (Item) Уголь 煤炭