Yes, you can filter water through charcoal, which is nearly the same as filtering water through activated carbon, just not as effective. Activated carbon is processed charcoal essentially.

Charcoal is the chunky black remnants of a wood fire. Look up "Charcoal" or "Lump Charcoal" on Google Images and you'll see a few pictures.

Ash is the dusty and flaky grey remains of a wood fire. Lye is made from wood ash and you can make soap with lye.

Other than that, whatever is left after a fire depends on what went into the fire, but basically charcoal and ash are the main remnants.