Charcoal is an alternative fuel source to Coal in Minecraft, and is extremely simple to manufacture for the creation of torches and fuel for furnaces.
Step One: Find Some Trees
The first ingredient for Charcoal is simply any type of Log in the game. Go out and do some lumberjacking, and once you have a healthy amount, haul it all back to your base.
A totally normal tree location. Totally.
Step Two: Acquire a Furnace
Now you need a Furnace. You can steal one from a Village, haul one from a mineshaft, or craft one. They're really cheap, just eight cobblestone after all.
Step Three: Making the Charcoal
So we have our Logs and we have our Furnace. Now it's as simple as cooking the logs in the furnace with any available fuel source. In my example, I just used (not at all suspiciously gathered) Coal to do the dirty work.
Upon completion, your logs will be turned into Charcoal:
Additional Notes
Charcoal is a carbon copy (no pun intended) of Coal in terms of statistics, and burn for the same duration (approximately smelting eight items per piece of Coal/Charcoal). Charcoal can also replace Coal in recipes for Torches, Soul Torches, Fire Charges, and Campfires. Note that Charcoal can not be crafted into blocks.
While this guide only provides information for vanilla, it should be noted that some mod recipes may also allow the substitution of Coal with Charcoal, or outright require it. Be sure to check your recipes, as it may be more cost-effective to use Charcoal in some cases! It is much easier to make more Charcoal than it is to locate and mine more Coal!