Morphisms of schemes
Recall that a scheme is a locally ringed space ${(X, {\mathscr O}_x)}$, i.e. a topological space and a sheaf of rings, such that for any ${x \in X}$ the stalk ${{\mathscr O}_{x,X}}$ is a local ring. Recall that the stalk is ${\left.\varprojlim_{U \ni x}\right. {\mathscr O}_X(U)}$.
An affine scheme is the scheme one gets from a commutative ring ${A}$ as follows:
$\displaystyle \begin{array}{l} X = \mathrm{Spec} A = \{\mathfrak{p} \subseteq A \textrm{, prime }\}\\ \quad \quad \quad \textrm{ with the base of topology } X_f = \{\mathfrak{p} \mid f \notin \mathfrak{p}\} \textrm{ where } f \in A\\ {\mathscr O}_X(X_f)=A_{(f)} \end{array} $
Let ${f: Y \rightarrow X}$ be a map of ringed spaces and ${{\mathscr F}}$ a sheaf on ${X}$. A pullback sheaf ${f^{-1}{\mathscr F}}$ is defined as follows:
$\displaystyle f^{-1}{\mathscr F}(U) = \varprojlim_{V \supseteq f(U)} {\mathscr F}(V) $
One has the natural map ${f^*: {\mathscr F} \rightarrow f^{-1}(F)}$. It is called inverse image map. One similarly defines ${f_*: Shv(X) \rightarrow Shv(Y)}$ and ${f^{-1}: \Gamma({\mathscr F},U) \rightarrow \Gamma(f_*({\mathscr F}),U)}$ for all ${U \subseteq X}$.
A morphism of locally ringed spaces is a morphism of underlying spaces ${f: Y \rightarrow X}$ and a morphism of sheaves ${f^*: f^{-1}{\mathscr O}_X \rightarrow {\mathscr O}_Y}$ (or ${f^*: {\mathscr O}_Y \rightarrow f_*{\mathscr O}_X}$), it is called local if the induced stalk morphisms map the maximal ideal to the maximal ideal.
Morphisms of affine schemes are induced by morphisms of rings (this is a theorem).
Non-local morphisms of ring spaces are not induced by ring morphisms. Example: a local ring and its field of fractions.
A morphism ${f: Y \rightarrow X}$ of schemes is called a closed immersion if it induces a homeomorphism between the underlying topological spaces of ${Y}$ and a closed subset of ${X}$ and is surjective on sections.
A scheme ${X}$ is called separated in the diagonal morphism ${\Delta: X \rightarrow X \times X}$ is a closed immersion (relative definition over some scheme ${S}$ is also possible).
Remark A category of schemes over a scheme ${S}$ in general does not have products, only fibred products, but it does have products if ${S}$ is separated.A category of schemes over a scheme ${S}$ in general does not have a final object, but it does have one if ${S}$ is affine.