Take a candle, light it, turn it on its side -- we all know what will happen. The convection cell that forms around the flame keeps licking up towards the sky (or ceiling) regardless of the orientation of the fuel.
But can the movement of air be overcome, can we make a fire burn horizontally? Well, the short answer is yes, yes we can -- we just have to use science.
The long answer
The chemical reaction that fire relies on is oxidation. While burning, part of the chemical building blocks of the piece of fuel are tied to atoms of O, releasing the energy they store in the initial chemical bonds as heat and light. The main by-products are carbon dioxide and water, most often in a gaseous form, mixed with other elements depending on the fuel used -- this is what creates smoke.
Now, the more physics-savvy of you already spotted two words in that previous paragraph, "gas" and "heat." When you heat a gas, or subject it to a strong enough electromagnetic field, you get plasma, one of the four fundamental states of matter we know of today.
The main difference between plasma and a gas is that in the former, a large fraction of the atoms are ionized -- because they environment is so hot, they slam into each other hard enough to allow some electrons to temporarily escape their host atom. Plasma is loosely described as an electrically neutral medium of unbound positive and negative particles (meaning that overall, it has zero electrical charge, but zoom in and it's more of a soup of +'s and -'s.)
Plasma thus gains some electrical properties that a non-ionized gas doesn’t have; it becomes conductive and it responds to electrical and magnetic fields. And that property is exactly what they guys from the Rino Foundation used to make their flame bend as it does in this video:
Fire is a genuine plasma, not the most ionized or homogeneous, but it is the kind we're most used to. Even small and relatively cool fires, like candle flames, respond to electric fields and are pretty conductive.
So, the next time you go to a party and really need to impress your crush with awesome fire-bending skills (and for some reason happen to carry a few sheets of copper and electrical wiring around, we don't judge), you'll know exactly how to go about doing that. You're welcome.
