An open subscheme of \(X\) is an open \(U \subseteq \abs{X}\) with \(\OO_U \da \ro{\OO_X}{U}\); every open subset of a scheme is a scheme in exactly one way.
A closed immersion \(i: Z \to X\) is a morphism which is a homeomorphism onto a closed subset and for which \(i^\sharp: \OO_X \to i_* \OO_Z\) is surjective. A closed subscheme is an equivalence class of closed immersions into \(X\).
Closed subschemes of \(\Spec A\) correspond exactly to ideals \(\mfa \normal A\), via \(Z = \Spec(A/\mfa)\). In general, closed subschemes of \(X\) correspond to quasicoherent ideal sheaves \(\mci \subseteq \OO_X\), with \(\OO_Z = \OO_X/\mci\).
(Closed subscheme sequences)
For a closed immersion \(i: Z \to X\) with ideal sheaf \(\mci\) there is an exact sequence of \(\OO_X\)-modules \[ 0 \to \mci \to \OO_X \to i_* \OO_Z \to 0 , \] and tensoring with a locally free \(\OO_X\)-module \(\mce\) keeps it exact: \(0 \to \mci \tensor \mce \to \mce \to i_*(i^* \mce) \to 0\). If \(Z = D\) is an effective Cartier divisor, then \(\mci = \OO_X(-D) = \OO_X(D)^\vee\), giving \[ 0 \to \OO_X(-D) \to \OO_X \to \OO_D \to 0 , \qquad 0 \to \mcl(-D) \to \mcl \to \ro{\mcl}{D} \to 0 \] for every invertible sheaf \(\mcl\). [@Har10a, Proposition II.6.18]
An open subset carries a unique induced scheme structure, while a closed subset can carry many closed-subscheme structures: \(V(x)\) and \(V(x^2)\) in \(\AA^1\) have the same support and different ideal sheaves.
For a closed immersion \(i\), the map on global sections \(\OO_X(X) \to \OO_Z(Z)\) need not be surjective: for \(Z=\ts{0,\infty}\subseteq\PP^1_k\) with its reduced structure, it is the diagonal \(k\to k\times k\).
A closed subscheme of an open subscheme is a locally closed subscheme; an immersion is a morphism that is an isomorphism onto a locally closed subscheme (D-MORIMM).