Throughout history, both coal and charcoal have been used as the fuel for forges across the world but charcoal sees more prevalence simply because trees are more common and easier to find than veins of coal.

There was an entire industry around charcoal making. These workers were called colliers in france and england. According to Prof Gerald Eggert at Penn State University, these guys would depopulate forests more than most any other proffession, cutting down ~300 hundred acres of wood each year just to keep a single blast furnace used in smelting iron running for that time. The trees were cut and stacked in massive, earth covered mounds that they would "burn" in an oxygen deprived atmosphere using coals from an existing fire (i won't describe the chemisty here, but i highly recommend you look into it) basically just letting it smolder for 2 weeks while monitoring it for any sign of flames to be quelled.

The process gave you almost pure carbon: charcoal.

The smith or smelter would buy charcoal from the collier, or maybe be involved in making it themselves, and to start a fire is as simple as doing it with regular wood. Gather your charcoal and using tinder and friction, or your flint and spark-rod, to slowly nurse your flames.

Once your flame is large enough, it is time to cover it with charcoal and start the bellows! Never forget the oft unappreciated bellows-tenders whose job in larger forges was to keep the fires lit but blowing large amounts of fresh air/oxygen into the flames so the fire can burn hotter. Without that airflow, you would never get the charcoal to burn hot enough to forge steel. The air increases the amount of fresh oxygen to burn, so by applying that to your fledgling fire you will engulf the charcoal in hot flames. Then just add more charcoal to keep it going. If you had coal instead, the process is exactly the same: start a normal fire then throw coal on it and blow air over the mound until it ignites.

Hope this helps.

You would probably not light your forge with a flint and tinder. That’s slow and fiddly work, and there’s a much easier way: borrow fire from your kitchen.

In societies that cook with fire (including medieval Western Europe), people typically kept fires going in their home hearth constantly. When you need the fire to cook, you add fuel and build a bed of coals. When cooking is done, you bank those coals to keep them warm until you need the fire again. The next time you want a fire, you add more fuel to those coals and bring the fire back. With a bit of experience, you can keep a fire going all year (in fact, many European cultures have a tradition of starting a new fire on a specific day of the year—a new fire for the new year, which might then be kept alive until the year’s end; this is for example celebrated on Samhain in early modern Ireland).

To light a charcoal forge (the most common fuel in medieval Europe, though there’s archaeological evidence that some medieval European smiths did use sea coal), all you have to do is place a hot coal beneath charcoal and blow air on it with your bellows. Most European smiths worked out of their homes, so the easiest way to start a fire in your forge would be to grab a coal from the kitchen hearth.

Medieval people did own flint and steel for starting fires. We find these in early medieval burials, and particularly in soldiers’ gear. When you’re away from home, you might need to make a fire from nothing. But when you were at home, you always had a fire near to hand.

Blacksmith and historian here. Flint and steel would be the primary method for striking a fire pre industrial revolution. Using the late medieval/early modern period as a frame of reference, the 15th century saw the invention of the arquebus, which used striker technology for ignition.

With flint and steel you get a spark, which you catch on charcloth and tinder. From there you'd put it into a small wood fire and surround the fire with your coals. From there you would provide steady oxygen flow from your double chamber pump bellows until the heart of your forge was roaring. This is the part of the process when the most smoke is generated as you burn off impurities. The smoke has a greenish/yellow hue from the sulphur. After this the forge is burning what's called coke.

A blacksmith forge is not like a charcoal grill or campfire where it's burning from bottom to top. Rather you have different layers. The heart of your forge is the part of the fire closest to the oxygen and is the hottest. It is your oxidizing environment, metal oxidizes (read: rusts with forge scale) in this section. The middle layer is where most of your forging is done. The top layer of actual burning material is your reducing environment where iron can actually take in carbon to become steel. obviously this is from a lay chemists understanding. Outside the reducing layer you have unlit coal. Coal is a great insulator. You can actually touch the coal and while it will be hot, it won't melt your skin. It's like touching a pan from the oven.