Informally speaking, the Casson invariant counts half the number of conjugacy classes of representations of the fundamental group of a homology 3-sphere M into the group SU(2).
The representation space of a compact oriented 3-manifold M is defined as \mathcal{R}(M)=R^{\mathrm{irr}}(M)/SU(2) where R^{\mathrm{irr}}(M) denotes the space of irreducible SU(2) representations of \pi_1 (M).