One of the applications that motivated the development of the theory is the solution of the Segal conjecture which states that the zeroth cohomotopy of a classifying space of a finite group is isomorphic to the completion of the Burnside ring. You can read about this in

• G. Carlsson. A survey of equivariant stable homotopy theory. Topology 31 (1992), 1-27.

There is also a book by J.P. May "Equivariant homotopy and cohomology theory" which discusses such applications.

Another application of equivariant cohomology is to the computation of ordinary cohomology of spaces with group actions. Equivariant cohomology has a nice property referred to as "localization", which allows to get cohomology information from just looking at fixed points. This can make computations of ring structures on equivariant cohomology easier than in ordinary cohomology (but one can then recover the ordinary cohomology ring from the equivariant one). You can read about this in the applications section of Tu's "What is ... equivariant cohomology?", the paper by Goresky-Kottwitz-MacPherson on equivariantly formal spaces, or the paper of Brion-Vergne on localization in equivariant cohomology

Finally, equivariant homotopy theory can also be used for classification of group actions on e.g. spheres. One can use equivariant versions of obstruction theory to classify maps between representation spheres of finite groups, cf. e.g.

• S.J. Willson, Equivariant maps between representation spheres. Pacific J. Math. 56, (1975), 291-296.

See also the works of tom Dieck, Petrie,... This application of equivariant homotopy theory is sort of a non-abelian version of representation theory.