Chemistry - What are the chemical reactions behind fire?
Solution 1:
Even though fire is one of the Greek classical elements, it is the only one that is not matter in our current understanding. What we experience as fire is the energy (in the form of light and heat) given off by the exothermic combustion of (usually) organic materials, like wood. The large amounts of thermal energy released by combustion often cause the gases in the fire to incandesce, that is, some of the kinetic energy produced by the fire is converted into electromagnetic radiation that we see (visible) and feel (infrared).
The chemical reaction for combustion is pretty simple. Take, for example, the combustion of n-octane $\ce{C8H18}$, which produces more than 5 MJ of energy per mole during combustion (from webbook.nist.gov).
$$\ce{2C8H18 +25O2->16CO2 +18H2O} \space\space \Delta_cH^o=-5430 \text{ kJ/mol} $$
The combustion of wood is more complex. The majority of the organic mass of dry dead wood is lignin and cellulose. Lignin is a highly-cross-linked copolymer of p-coumaryl alcohol, $\ce{C9H10O2}$, coniferyl alcohol, $\ce{C10H12O3}$, and sinapyl alcohol, $\ce{C11H14O4}$. Cellulose is a linear polymer of glucose, with formula $\ce{(C6H10O5)}_n$.
Different species will have different ratios of lignin to cellulose, different cross-link densities in the lignin, and different ratios of coumaryl to coniferyl to sinapyl alcohols. The formula for lignin could be expressed then as $\ce{(C9H10O2)}_x \cdot \ce{(C10H12O3)}_y \cdot \ce{(C11H14O4)}_z$.
The equation for the combustion reaction for cellulose is: $$\ce{(C6H10O5)}_n +6n\ce{O2->}+6n\ce{CO2}+5n\ce{H2O} $$
Since lignin is more complex, its combustion equation is more complex: $$2[\ce{(C9H10O2)}_x \cdot \ce{(C10H12O3)}_y \cdot \ce{(C11H14O4)}_z]+(21x+23y+25z)\ce{O2}\\ \ce{->}(18x+20y+22z)\ce{CO2}+(10x+12y+14z)\ce{H2O}$$
And assuming the variable composition of wood as a ratio (A:B) of cellulose to lignin, the overall combustion reaction becomes the following monstrosity:
$$A\ce{(C6H10O5)}_n +2B[\ce{(C9H10O2)}_x \cdot \ce{(C10H12O3)}_y \cdot \ce{(C11H14O4)}_z]+[6An+B(21x+23y+25z)]\ce{O2}\\ \ce{->}[6An+B (18x+20y+22z)]\ce{CO2}+[5An+B(10x+12y+14z)]\ce{H2O}$$
Given the number of variables ($A, B, n, x, y, z$), the $\Delta_cH^o$ for this reaction is difficult to determine (but not impossible, assuming we know some thermochemical reference data). Whatever its value, it is exothermic enough to induce incandescence.
Solution 2:
The chemical reactions of a "typical fire" using wood as the inflammable matter are extremely complex and not known to their full extend. The wood isn't actually burning but gasified (mostly superficial) components of the wood, which yield a plasma in an environment that provides sufficient excitation energy. Accelerants are often used to this end.
The situation is aggravated by the many radical species which are produced in chain reactions and the temperature gradient of the flame which favors different species.
The form of fire depends on many parameters, gravity included...
A search for "estimating the number of chemical species in a wood fire" brings up interesting literature which may serve you as a starting point, in addition to Ben Norris's answer.
Solution 3:
Ben's answer covers it well. There is another aspect which affects the products of fire; that is the temperature of the fire, which varies according to the phase of the fire, among other things.
I found the following list in some educational materials for fire fighters:
Temperature Reaction
3920F Production of water vapor, carbon dioxide, formic and (2000C) acetic acids
3920-5360F Less water vapor - some carbon monoxide - still primarily an (2000-2800C) endothermic reaction (absorbing heat)
5360-9320F Exothermic reaction (giving off heat) with flammable vapors (2800-5000C) and particulates; some secondary reaction from charcoal formed
Over 9320F Residue primarily charcoal with notable catalytic action (5000C)
Fire is fascinating. As already noted in answers above, fire is a mixture of matter and energy. Fire consists of fuel, oxygen (or another oxidizer) and heat interacting in a chain reaction yielding more heat and oxidized byproducts.