The free rider problem exists when people enjoy the benefits of government provided goods independent of whether they pay for them.
In the analyses of economics and political science, free riders are actors who take more than their fair share of the benefits or do not shoulder their fair share of the costs of their use of a resource. The free rider problem is the question of how to prevent free riding from taking place, or at least limit its effects.
Because the notion of "fairness" is highly subjective, free riding is usually only considered to be an economic "problem" when it leads to the non-production or under-production of a public good, and thus to Pareto inefficiency, or when it leads to the excessive use of a common property resource.
The classic example is national defense; I am protected whether or not I pay, so there's no reason for me to pay unless forced to do so. In such cases, the government must provide the good itself and force individuals to pay for it with taxes.
Source: R A Musgrave and P B Musgrave, Public Finance in Theory and Practice (New York, 1975)