How can a Pigovian tax resolve a market failure that occurs when consumers' consumption causes a negative exernality?
A Pigovian tax would resolve the market failure by decreasing the quantity demanded (assuming that the tax falls on consumers) of the product whose use is causing the negative externality. Please follow this link to read a proposal for Pigovian taxes by one of America's leading professors of Economics.
With a gasoline tax like the one Mankiw is promoting in the link above, the government would in effect be raising the price of gasoline. This would cause a move to the left along consumers' demand curve for gasoline. The move to the left would cause a new equilibrium quantity that would be lower than the equilibrium quantity would have been without the Pigovian tax.
Thus, by raising prices, Pigovian taxes discourage the consumption that leads to negative externalities.