A forge is pretty simple. You need two main components.
A container for your "fire".
A method of feeding air into your fire.
The rest ends up being ... creative design.
Most coal or charcoal forges are bottom draft - they feed the air up from the bottom. But side-draft is also an option - the air is fed in from the side.
And the body of the forge is just some sort of bowl/trough/dish to hold your fuel in place. A tin "washtub" works, but is kind of thin metal - so you line it with clay or firebrick. This also helps hold/retain/reflect your heat.
But many people choose something a little thicker/heavier to start with. A brake drum from a truck works well. It is thick/heavy, is already dished, and comes with a hole through the center bottom.
Then just get some large pipe pieces from the store to bolt onto the outside of that brake drum - to feed the air up into the fire. Needed pieces? A T, a flange, an end plug, and two short/long sections of pipe. The flange bolts to the brake drum. Then screw in a short section of pipe. Now thread on that T coupling. Add that other long section of pipe off of the side of that T coupling - this is where your air supply will come in. Use that end plug to close up the other end of the T on the bottom. That plug allows you to clean out any ash that falls down into your pipes. On top of the hole inside your brake drum, bolt on a cast iron floor drain cover, or a piece of heavy steel plate with a lot of holes drilled through it. You can bolt on legs to that brake drum if you want to, or just set it up on top of a bunch of stacked up bricks.
An air supply is as close as that cheapo hair blow-dryer. Figure out a way to attach it to that long section of pipe sticking out from under your brake drum. Duct Tape works well - especially if that pipe coming out is fairly long to cut down on the heat radiating out of your forge. If you don't have regular 110 volt available, just go to an auto salvage yard and get the car heater blower from a junker and run it off of a 12 volt car battery.
Clay or refractory cement inside your forge does help even out and reflect the heat, and it also helps prolong the life of your outside "container". But you can also just use clay from the yard/field. Or mix up a little cement and put that in. Yes the clay and cement will crack and eventually crumble. But they are cheap enough to patch or just chip out and replace.
So something to contain your fire/fuel, and a method to get extra air into your fire. That's all you really need. The rest is just ... creative design.
Mikey - that grumpy ol' German blacksmith out in the Hinterlands