Government failure is a situation in which the government intervenes to correct for externalities and ends up making things worse. Since government failures are pervasive in society, politicians usually have to choose between government failure and market failure.
For example, let's say there is air pollution (a negative externality), and government decides to make a subsidy for corporations that do not put chemicals into the air. Then, all the factories that were putting chemicals into the air instead add tons of toxic material to the water supply, destroying many sources of drinking water, and causing the death of some endangered species of fish. That would be an example of government failure.