A Pigovian tax is a tax on an activity that generates a negative externality. They are named after economist Arthur Pigou, who is best known for his work in developing the theory of externalities.
Theoretically, these taxes should correct the negative externality by perfectly offsetting all the negative effects, guaranteeing that the people producing these negative externalities are willing to pay all costs of their actions. These costs are necessarily passed on to the consumer, so people only consume these goods/services if they are willing to pay the true cost.
Common examples are taxes on industries that pollute, products that harm consumer health, and inefficient land use. While there is much debate around Pigovian taxes, perhaps one of the most serious problems is simply the measurement of the externalities involved. For example, there has long been significant debate over the causes and costs of global warming, and it is trivially true that no current system -- Pigovian or otherwise -- is responding optimally to the situation.
There is also a deeper issue in that Pigovian taxes are intended to be efficient, but are not necessarily fair. That is, if you live next door to a hog farm and suffer ill health from airborne particulate waste, Pigovian taxes should ideally ensure that the producer and the consumers are paying for harming you, but are not guaranteed, or intended, to put money in your pocket to pay for your medical bills. This is a matter for the legal system, specifically tort liability laws; how exactly a system can best balance Pigovian taxes and tort law is a matter of some debate.
The flip side of the Pigovian tax is a Pigovian subsidy, in which goods and services that are under-supplied are subsidized by the government.