I am confused.

I've seen, flames, flame particles, and flame forges.

So what exactly is needed for each of them and/or how do they work?

Flames are the items you see ingame equipped to a player that have that 'flamelike' look.

Flame particles is what you can equip to a flame, to change the particles shape and color of the actual flame.
Flame forgers arepeople who are skilled in making unique styled flames, a flame forge is in the version tb 3.9 and later, this lets you create flames after playing around with different settings.

So, you add the flame particle texture onto the flame.

But how does the flame forge work?

Do you equip the forge onto the flame as well?

nono, the flame forger is a utility in the toribash game menu.
You can find it by going to setup > utilities > Flame forger. As I said this is only in version 3.9 and further.

Basically the flame forger menu has options on it which you can modify, and a preview of your tori with the flame as you change it with the options.

When you are ready and click the 'forge' button it takes you to the forums, where you are given the price of the flame and if you still want to/can forge it. The flame is created in your inventory then.

I see, so you need neither flame nor particles for it?

Exactly. It's just a thing you use to create the actual flame