A morphism of schemes f : X \to Y consists of a continuous map f and, for every open set V \subset Y, a ring homomorphism \mathcal{O}_Y(V) \to \mathcal{O}_X(f^{-1}(V)) that pulls functions back and commutes with restriction. On every stalk the induced map \mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x} must be a local homomorphism, sending the maximal ideal into the maximal ideal [1]. In other words, a function vanishes at f(x) exactly when its pullback vanishes at x.
Between affine schemes these morphisms are the ring homomorphisms. A ring homomorphism \varphi : A \to B gives a morphism \operatorname{Spec} B \to \operatorname{Spec} A that sends \mathfrak{q} to \varphi^{-1}(\mathfrak{q}), and every morphism \operatorname{Spec} B \to \operatorname{Spec} A comes from exactly one \varphi. More generally, morphisms from any scheme X to \operatorname{Spec} A correspond to ring homomorphisms A \to \mathcal{O}_X(X) [1].
The homomorphism \mathbb{C}[x] \to \mathbb{C}[t], x \mapsto t^2, gives the morphism t \mapsto t^2 of complex lines. Over x = a with a \neq 0 lie the two points \sqrt{a} and -\sqrt{a}, and over x = 0 lies only t = 0, although the fiber \operatorname{Spec} \mathbb{C}[t]/(t^2) still has length 2. In the figure the angles double and the distances from the center are squared, so opposite rays come together on one ray, and only the origin is reached from a single point.
Let K be a field. A morphism \operatorname{Spec} K \to X sends the single point to some x \in X, and the local condition forces the maximal ideal \mathfrak{m}_x of \mathcal{O}_{X,x} to go to 0, so the morphism factors through the residue field \mathcal{O}_{X,x}/\mathfrak{m}_x. Morphisms \operatorname{Spec} K \to X therefore correspond to a point x together with an embedding of its residue field into K [1].
For X = \operatorname{Spec} k[x] and K = k(t), a morphism is given by x \mapsto r(t) for a rational function r. If r = a is constant, the kernel of k[x] \to k(t) is (x - a) and the point goes to the closed point x = a. If r is not constant, it is transcendental over k, no nonzero polynomial p has p(r) = 0, the kernel is (0), and the point goes to the generic point. In the figure the value of a constant r is the root a of p = (x - a)(x - c), so p(r) = 0; once the graph moves, its values fill an interval, and no nonzero polynomial vanishes at all of them.
The local condition is what rules out the other placement. The inclusion \mathbb{Z}_{(p)} \to \mathbb{Q}, together with the map that sends the point of \operatorname{Spec} \mathbb{Q} to the closed point of \operatorname{Spec} \mathbb{Z}_{(p)}, still defines a morphism of ringed spaces. On the stalk at the closed point, however, it is \mathbb{Z}_{(p)} \to \mathbb{Q}, which sends p, an element of the maximal ideal, to a unit, so it is not local, and the only morphism of schemes sends the point to the generic point [1].
Over \mathbb{F}_p the homomorphism \mathbb{F}_p[x] \to \mathbb{F}_p[x], g \mapsto g^p, sends x to x^p and defines a morphism F from \mathbb{A}^1 = \operatorname{Spec} \mathbb{F}_p[x] to itself. It fixes every point, since g^p \in \mathfrak{q} exactly when g \in \mathfrak{q} for a prime ideal \mathfrak{q}. Yet F is not an isomorphism, because the image of \mathbb{F}_p[x] is the smaller ring \mathbb{F}_p[x^p] [2].
The points over the algebraic closure \bar{\mathbb{F}}_p do move. A closed point of \mathbb{A}^1 is an irreducible polynomial, and its roots in \bar{\mathbb{F}}_p form one orbit of \alpha \mapsto \alpha^p, whose size is the degree. In the figure the closed points of degree 1 to 4 over \mathbb{F}_2 are drawn as polygons with the roots at the corners; the Frobenius moves every root one corner forward, and every polygon stays where it is.
So a morphism of schemes is not determined by what it does to points. The rest of the information is in the pullback of functions, here the difference between x and x^p, which take the same value at every point of \mathbb{A}^1(\mathbb{F}_p) and are still different functions. Tangent vectors show the same thing: morphisms from \operatorname{Spec} k[\varepsilon]/(\varepsilon^2), a scheme with one point, to a scheme X over k correspond to points of X with residue field k together with a tangent vector there [1].