Why do some materials turn black and others white after combustion?
The objects which turn black are usually carbon based/organic compounds. When burned/combusted they follow one of the following unbalanced reactions:
\begin{align}
\ce C_x(\ce H_y\ce O_z)_\text{(solid/liquid)} + {\ce{O2}}_\text{(gas)} &amp;\ce{-&gt;} \ce C_\text{(solid)} + \ce{H_2O}_\text{(liquid)} + {\ce{CO_2}}_\text{(gas)}\\
\ce C_x(\ce H_y\ce O_z)_\text{(solid/liquid)} + {\ce{O2}}_\text{(gas)} &amp;\ce{-&gt;}  \ce{H2O}_\text{(liquid)} + {\ce{CO2}}_\text{(gas)}\\
\ce C_x(\ce H_y\ce O_z)_\text{(solid/liquid)} + {\ce{O2}}_\text{(gas)} &amp;\ce{-&gt;}  \ce {H2O}_\text{(liquid)} + {\ce{CO2}}_\text{(gas)} + \ce{CO}_\text{(gas)}
\end{align}

You'll notice that some of the reactions will produce solid carbon, which is the black thing you see.

On the other hand, inorganic compounds begin to decompose into their ions and/or release their  "trapped" water molecules. As you'll notice, most ionic compounds are white solid salts.