A general answer, sure a chemist could give a more complete/detailed answer.

The black stuff isn't ash, it's soot, made up mostly of unburned carbon (giving the black colour). Ash is pale grey-white and consists of non-combustible compounds. When you get charred wood, for example, the black is carbon that hasn't burned yet, and if you let it continue to burn, the black carbon will eventually burn away and leave behind the pale ash. The black dust that you may be thinking of is just fine particles of soot, which are a component of smoke, drawn up by the heat and release of other gasses which make up the smoke.