Let \(Z \subseteq \abs{X}\) be closed. For \(X = \Spec A\) take \(\mfa \da \Intersect_{\mfp \in Z} \mfp\), a radical ideal, and give \(Z\) the structure \(\Spec(A/\mfa)\). These agree on overlaps, so they glue to a closed subscheme structure on \(Z\) for any \(X\): the reduced induced structure \(Z^\red\).
\(Z^\red\) is the unique reduced closed subscheme of \(X\) with underlying space \(Z\), and it is the smallest closed subscheme with that space: every closed subscheme supported on \(Z\) contains \(Z^\red\) as a closed subscheme. Every morphism \(T\to X\) from a reduced scheme \(T\) whose image lies in \(Z\) factors uniquely through \(Z^\red\).
Taking \(Z = \abs{X}\) gives the reduction \(X^\red \to X\), a homeomorphism that is an isomorphism if and only if \(X\) is reduced. For an ideal \(\mfa\subseteq A\), \(\bigcap_{\mfp\supseteq\mfa}\mfp=\sqrt{\mfa}\), so the reduced induced structure on \(V(\mfa)\subseteq\Spec A\) is \(\Spec(A/\sqrt{\mfa})\).