Differential inequalities occurring in problems of obstacle contact problems are recast into variational inequalities and analyzed by finite element methods. A new a posteriori error estimator, which is essential in adaptive finite element method, is introduced to capture the errors in finite element approximations of these variational inequalities. In order to construct a posteriori error estimates, saddle point problems are introduced using Lagrange parameters and upper bounds are provided. The global upper bound is localized by a special mixed formulation, which leads to upper bounds of the element errors. A numerical experiment is performed on an obstacle contact problem to check the effectivity index both in a local and a global sense.