What are the chemical reactions behind fire?
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). 

$$\ce{2C8H18 +25O2-&gt;16CO2 +18H2O} \space\space \Delta_cH^o=-5430 \text{ kJ/mol} $$

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:
$$\ce{(C6H10O5)}_n +6n\ce{O2-&gt;}+6n\ce{CO2}+5n\ce{H2O} $$

Since lignin is more complex, its combustion equation is more complex:
$$2[\ce{(C9H10O2)}_x \cdot \ce{(C10H12O3)}_y \cdot \ce{(C11H14O4)}_z]+(21x+23y+25z)\ce{O2}\\ \ce{-&gt;}(18x+20y+22z)\ce{CO2}+(10x+12y+14z)\ce{H2O}$$

And assuming the variable composition of wood as a ratio (A:B) of cellulose to lignin, the overall combustion reaction becomes the following monstrosity:

$$A\ce{(C6H10O5)}_n +2B[\ce{(C9H10O2)}_x \cdot \ce{(C10H12O3)}_y \cdot \ce{(C11H14O4)}_z]+[6An+B(21x+23y+25z)]\ce{O2}\\ \ce{-&gt;}[6An+B (18x+20y+22z)]\ce{CO2}+[5An+B(10x+12y+14z)]\ce{H2O}$$

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.