Elaborating on the first sentence of Harry's answer - here's another motivation for considering nilpotents. The scheme $D=Spec(K[t]/(t^2))$ set-theoretically consists of a single point, but this point has a non-trivial (one-dimensional) tangent space with a distinguished non-zero tangent vector. For a scheme $X/K$, a map $D\to X$ amounts to a point $x\in X$ and an element of the tangent space $T_x(X)$.
This observations observation is used to characterize tangent spaces as fibers of the maps $X(D)\to X(Spec(K))$. This is useful in the study of Lie algebras of algebraic groups, for example.
