It depends - if the charcoal was the briquettes supplied for barbecue use, these often are impregnated with petrochemicals to make them burn better, so don't put on the garden. If its naturally produced charcoal from wood, with no further treatment, then you can use it sparingly on the ground.

Bonfire ash, depends what you've been burning - if its just wood from the garden, you can spread that around the soil, but if you use too much, it can have an alkalysing effect on the soil ph.

Ash from old painted wood can also have toxic metals from the paint.

Wood ash contains, quite literally, potash - good source of potassium but in an alkaline form that used to be called lye. Fine for plants that tolerate high pH or adding to peat based composts. Huge quantities of ash were used to grow tomatoes in the past.
Charcoal (homemade from wood) has a slight purifying properties as it can adsorb certain organics. Biochar is a new buzz word, basically black carbon from plants, and cheap potting composts are full of charcoal/biochar. But to be effective it must be ground fine, lumps of 5mm or more will just sit in the soil doing nothing for years.

For articles on biochar, google "terra prieta do indio"