morphism of varieties
If varieties are the objects of algebraic geometry, morphisms are the allowed motions between them: maps that respect the polynomial structure. Just as continuous maps are the right maps between topological spaces and linear maps between vector spaces, morphisms are the maps that 'algebraic geometry can see'.
A morphism from a variety X to a variety Y is a map of the underlying spaces that is continuous for the Zariski topology and pulls back regular functions to regular functions: for every open V in Y and every regular g on V, the composite g composed with the map is regular on its preimage. For affine varieties this is concrete — a morphism X to A^m is just an m-tuple of regular functions on X, and a morphism X to Y in A^m is such a tuple landing inside Y.
The deep fact is the equivalence with algebra. Over a fixed field, sending a variety to its coordinate ring is a contravariant equivalence between affine varieties and finitely generated reduced k-algebras, and morphisms X to Y correspond exactly to k-algebra homomorphisms k[Y] to k[X]. A morphism that has a two-sided inverse morphism is an isomorphism, matching an isomorphism of coordinate rings; note that a bijective morphism need not be an isomorphism, as the Frobenius and the cuspidal-cubic normalization show.
The map t maps to (t^2, t^3) is a morphism A^1 to the cuspidal cubic V(y^2 - x^3) in A^2. It is a bijection, yet not an isomorphism: the induced map k[x, y]/(y^2 - x^3) to k[t] is the inclusion onto the subring k[t^2, t^3], which is not all of k[t].
A bijective morphism that fails to be an isomorphism — the cusp is the obstruction.