Coal

Coal lumps are items mainly obtained by harvesting coal ore blocks.

Charcoal is an alternative to coal. It can be used for most of the same things coal can, except in creating coal blocks.

Aside from coal having the ability to craft blocks, coal and charcoal are functionally identical; except for the fact that they cannot stack together.

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

Mining
Coal ore mined using a pickaxe will drop one unit of coal. If the pickaxe is enchanted with fortune, it may drop an extra unit per level of fortune, up to a maximum of 4 with Fortune III.

Wither Skeletons
A Wither Skeleton may drop a coal lump upon death.

Natural generation
Coal can be found in stacks of 3 to 8 in the chests in stronghold store rooms and in minecart with chests in abandoned mine shafts.

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

Coal and charcoal are also the only fuels accepted by powered minecarts. They provide approximately four minutes of transit each.

In Pocket Edition, in Survival, if you put charcoal in a furnace you may be able to stack coal on the charcoal, and the coal with automatically change into charcoal. This also works backwards.

Video
Note: The video does not show a crafting recipe for Blocks of Coal. This was before Blocks of Coal were added in 1.6.1.

Note: The video states that charcoal looks the same as coal. This was before 1.6.1, when charcoal's texture was changed.