We investigate eigenvalues of the zero-divisor graph $Γ(R)$ of finite commutative rings $R$ and study the interplay between these eigenvalues, the ring-theoretic properties of $R$ and the graph-theoretic properties of $Γ(R)$. The graph $Γ(R)$ is defined as the graph with vertex set consisting of all non-zero zero-divisors of $R$ and adjacent vertices $x,y$ whenever $xy = 0$. We provide formulas for the nullity of $Γ(R)$, i.e. the multiplicity of the eigenvalue 0 of $Γ(R)$. Moreover, we precisely determine the spectra of $Γ(\mathbb Z_p \times \mathbb Z_p \times \mathbb Z_p)$ and $Γ(\mathbb Z_p \times \mathbb Z_p \times \mathbb Z_p \times \mathbb Z_p)$ for a prime number $p$. We introduce a graph product $\times_Γ$ with the property that $Γ(R) \cong Γ(R_1) \times_Γ \ldots \times_Γ Γ(R_r)$ whenever $R \cong R_1 \times \ldots \times R_r.$ With this product, we find relations between the number of vertices of the zero-divisor graph $Γ(R)$, the compressed zero-divisor graph, the structure of the ring $R$ and the eigenvalues of $Γ(R)$.