Let \(R\) be a commutative ring. The Jacobson radical of \(R\) is the intersection of the annihilators of the simple \(R\)-modules: \[ J(R)=\bigcap_{\mfm\in\mspec R}\mfm=\bigcap_{M\text{ simple}}\Ann_R(M). \] If \(R\) is a finitely generated algebra over a field \(k\), then \(J(R)\) equals the nilradical \(\nilrad{R}\).
A simple module \(M\) is generated by any nonzero \(m\in M\), so \(M\cong R/\Ann_R(m)\), and simplicity makes \(\Ann_R(m)\) a maximal ideal \(\mfm\); then \(\Ann_R(M)=\mfm\). Conversely \(R/\mfm\) is simple with annihilator \(\mfm\) for every maximal ideal \(\mfm\). So the two intersections run over the same ideals.
For a finitely generated \(k\)-algebra, every prime ideal is an intersection of maximal ideals by the Nullstellensatz, so the intersection of all maximal ideals equals the intersection of all prime ideals, which is \(\nilrad{R}\).
Write \(R=k[x_1,\ldots,x_n]/I\). Then \(J(R)=\nilrad{R}=\sqrt I/I\), so \(J(R)=0\) exactly when \(I\) is a radical ideal; for \(k\) algebraically closed, these are the ideals \(I(Z)\) of affine algebraic sets \(Z\subseteq\AA^n_k\), whose coordinate rings \(k[x_1,\ldots,x_n]/I(Z)\) are the rings of regular functions on \(Z\).