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 quite efficient as a source of fuel for Furnaces, being able to smelt eight units of ore per lump of coal (hence eight lumps will smelt a full stack of 64 ore units). 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 64 lumps 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 yet more coal as you dig). Perhaps the best alternative to coal-based lighting would be Netherrack, which, if manually ignited, will burn forever.

Coal can be found anywhere in the map, similar to Iron ore. However, it is the most abundant resource, and can be found in veins of 1-40, or more.

Charcoal
Charcoal is an alternative to coal. Smelting Wood in a furnace creates Charcoal, which can be substituted for Coal in all uses, 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, as no coal needs to be mined in order to craft torches any more. Charcoal does not stack with Coal. Charcoal and Coal look exactly the same and cannot be changed with a texture pack as the texture is shared between the two.

Also if you are playing on survival (especially on Hard) then you can just make 1 lump of charcoal then use that on 8 wood block to turn 1 lump into 8 again making it easier to survive by multiplying the amount of torches that can be made and/or amount of items you can smelt.

Smelting
Charcoal is also used for making torches

Efficiency
It is more efficient, though time-consuming, to turn wood into charcoal for smelting than to use wood for smelting. Ordinarily, a wood block smelts six items if broken into planks: each wood block makes four wooden planks, and each wooden plank smelts 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.

Therefore, it is more efficient to take seven wood blocks, turn one wood block into planks, and use those four planks to smelt the other six unbroken wood blocks into charcoal. Those six charcoal can then smelt 48 items. Ordinarily, seven wood blocks would only smelt 42 items, since seven wood blocks makes 28 planks. This process thus gives a 14% smelting increase over using planks for fuel. It does, however, require waiting for the six wood blocks to become charcoal.

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 there is even more time required for smelting the wood.

Other methods of creating Charcoal without coal:

Use one wood block -> Planks -> 1 set of sticks(4) use 2 of the sticks to smelt your remaining raw wood, this will give you one piece of charcoal, and you can use that to smelt.

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 cohesively with each other, even though they can be used interchangeably.

Data values
Kohle (Item) Charbon (objet) Szén (tárgy) Kool (voorwerp) Węgiel Уголь 煤炭