the functor of points
When a scheme is defined as a locally ringed space glued from spectra, its 'points' are prime ideals — a rather forbidding and rigid notion. The functor-of-points perspective offers a completely different, often more intuitive way to understand a scheme X: instead of asking 'what are the points of X?', ask 'what are the maps INTO X from every other scheme?'. The collection of all such maps, organized as a functor, determines X completely and frequently matches the naive idea of solutions to equations far better than prime ideals do.
Precisely, a scheme X gives rise to a contravariant functor h_X from schemes to sets, sending each scheme T to the set h_X(T) = Hom(T, X) of all morphisms from T to X, called the T-valued points of X. By the Yoneda lemma this functor remembers X completely: X is recovered (up to isomorphism) from h_X, and a morphism X -> Y is the same as a natural transformation h_X -> h_Y. When X = Spec A is affine and T = Spec R is affine, the anti-equivalence makes T-valued points concrete: Hom(Spec R, Spec A) = Hom_rings(A, R), the ring homomorphisms A -> R. So the R-points of the scheme cut out by polynomials f_1, ..., f_m are exactly the simultaneous solutions of those polynomials with coordinates in R — the elementary meaning of 'points' is restored, but now allowed in EVERY ring R at once. A functor from rings to sets that arises this way (is isomorphic to some h_X) is called representable, represented by X.
This viewpoint is the practical heart of moduli theory and modern algebraic geometry: one defines a space of geometric objects (curves of given genus, vector bundles, subschemes) by writing down the functor 'families of such objects over T' and then asking whether it is representable by a scheme. It also clarifies extra structure — a group scheme is just a scheme whose functor of points lands in groups, like GL_n with GL_n(R) = invertible n-by-n matrices over R. Two honest cautions. First, you must let T range over ALL schemes (or all rings), not just fields: restricting to k-points loses information, and notably the nilpotents in non-reduced T are exactly what detect tangent vectors and infinitesimal deformations — the dual numbers k[epsilon]/(epsilon^2) probe tangent spaces. Second, not every functor is representable; many natural moduli functors are only representable after enlarging the category (to algebraic spaces or stacks), so 'write down the functor' does not by itself guarantee a scheme exists.
The tangent space at a point trick: for a k-scheme X and a k-point x, the X-valued points of the dual numbers — maps Spec k[epsilon]/(epsilon^2) -> X centered at x — are exactly the tangent vectors of X at x. So 'infinitesimal points' (using a non-reduced test scheme) literally compute the Zariski tangent space, something invisible if you only allow field-valued points.
Dual-number points Spec k[epsilon]/(epsilon^2) -> X are tangent vectors: the functor sees infinitesimals.
T must range over all schemes (all rings), not just fields — the nilpotents in non-reduced test schemes are exactly what detect tangent vectors and deformations. And not every moduli functor is representable by a scheme; some need stacks.