Burns (cigarettes or other)
Penetrating stains or wax products
Use a damp cloth to moisten the area first, then gently rub with steel wool or very fine sand paper - preferably the wet & dry variety. For deeper burns the damaged wood will need to be carefully removed with a knife.
Lacquer or other surface finish
Surface burns can be renovated using a wood flooring kit designed for lacquer/urethane floor finishes.