  Let $C_2$ denote the cyclic group of order two. Given a manifold with a
$C_2$-action, we can consider its equivariant Bredon $RO(C_2)$-graded
cohomology. In this paper, we develop a theory of fundamental classes for
equivariant submanifolds in $RO(C_2)$-graded cohomology in constant
$\mathbb{Z}/2$ coefficients. We show the cohomology of any $C_2$-surface is
generated by fundamental classes, and these classes can be used to easily
compute the ring structure. To define fundamental classes we are led to study
the cohomology of Thom spaces of equivariant vector bundles. In general the
cohomology of the Thom space is not just a shift of the cohomology of the base
space, but we show there are still elements that act as Thom classes, and
cupping with these classes gives an isomorphism within a certain range.
