This answer is biased towards the relation to Group Cohomology, as that's where I first learned about equivariant cohomology while studying under Ken Brown:

1. Quillen's famous The Spectrum of an Equivariant Cohomology Ring I+II (which proves that the Krull dimension of $H^*(G,\mathbb{Z}_p)$ is the p-rank, i.e. maximum rank of an elementary abelian p-subgroup of $G$, and that the minimal prime ideals of the ring are in 1-1 correspondence with the conjugacy classes of maximal elementary aabelian p-subgroups).
2. Atiyah & Bott's The Moment Map and Equivariant Cohomology (on localization, but also just a nice expository paper).
3. Duflot's famous Depth and Equivariant Cohomology (which proves that the depth of the equivariant cohomology ring $H^*_G(X;\mathbb{Z}_p)$ is at least the maximum rank of a central p-torus acting trivially on the space $X$).
   These are the three main references I'd give, but others include:
4. Duflot's Localizations of Equivariant Cohomology Rings (which computes the localization of the equivariant cohomology ring localized at one of its minimal prime ideals).
5. Duflot's The Associated Primes of $H^*_G(X)$ (which proves that the associated primes of $H^*_G(X;\mathbb{Z}_p)$ are invariant under Steenrod operations, and in fact can be obtained by restricting the ring to that of a p-torus).
6. Adem's Torsion in Equivariant Cohomology (self-explanatory).

And perhaps just as a meal-time reading: Loring Tu's expository article What is Equivariant Cohomology? in AMS Notices.

This answer is biased towards the relation to Group Cohomology, as that's where I first learned about equivariant cohomology while studying under Ken Brown:

1. Quillen's famous The Spectrum of an Equivariant Cohomology Ring I+II (which proves that the Krull dimension of $H^*(G,\mathbb{Z}_p)$ is the p-rank, i.e. maximum rank of an elementary abelian p-subgroup of $G$, and that the minimal prime ideals of the ring are in 1-1 correspondence with the conjugacy classes of maximal elementary abelian p-subgroups).
2. Atiyah & Bott's The Moment Map and Equivariant Cohomology (on localization, but also just a nice expository paper).
3. Duflot's famous Depth and Equivariant Cohomology (which proves that the depth of the equivariant cohomology ring $H^*_G(X;\mathbb{Z}_p)$ is at least the maximum rank of a central p-torus acting trivially on the space $X$).
   These are the three main references I'd give, but others include:
4. Duflot's Localizations of Equivariant Cohomology Rings (which computes the localization of the equivariant cohomology ring localized at one of its minimal prime ideals).
5. Duflot's The Associated Primes of $H^*_G(X)$ (which proves that the associated primes of $H^*_G(X;\mathbb{Z}_p)$ are invariant under Steenrod operations, and in fact can be obtained by restricting the ring to that of a p-torus).
6. Adem's Torsion in Equivariant Cohomology (self-explanatory).

And perhaps just as a meal-time reading: Loring Tu's expository article What is Equivariant Cohomology? in AMS Notices.