a birational map
Two varieties can be 'the same except along a thin patch you are allowed to ignore.' A birational map is the precise notion of such a near-isomorphism: an invertible correspondence that is defined and bijective away from a lower-dimensional bad set on each side. The blow-up of the plane at a point and the plane itself are birational — they agree everywhere except over that one point, which the blow-up has replaced by a whole line.
A rational map f: X --> Y between irreducible varieties is given by rational functions; it need only be defined on a dense open subset. It is dominant if its image is dense. A birational map is a rational map f: X --> Y that admits a rational inverse g: Y --> X with f composed g and g composed f equal to the identity wherever both sides are defined. Equivalently, and this is the clean characterization, X and Y are birational exactly when their function fields k(X) and k(Y) are isomorphic as k-algebras. A birational map restricts to an honest isomorphism between two dense open subsets U in X and V in Y; the difference between X and Y is concentrated on the complementary closed sets, which have strictly smaller dimension.
Birational classification is one of the central programs of algebraic geometry — classifying varieties up to birational equivalence is far coarser, and far more tractable, than up to isomorphism, and resolution of singularities and the minimal model program live here. A variety is called rational if it is birational to projective space, meaning k(X) is purely transcendental over k. Honest caveat: birational varieties are genuinely different as varieties. They share dimension and the function field but can differ in their singularities, their cohomology beyond birational invariants, and even in being smooth or not — so 'birational' must never be upgraded to 'isomorphic.'
The map P^1 --> the nodal cubic y^2 = x^2(x + 1), sending t to (t^2 - 1, t(t^2 - 1)), is birational: it is an isomorphism away from the node, where the two values t = 1 and t = -1 both map to the origin. So the singular cubic is rational — birational to a line — even though it is not isomorphic to one.
Birational equals isomorphic on dense opens; the difference hides on lower-dimensional sets, here a single node.
Birational does not mean isomorphic. The two varieties agree only on dense open subsets and can genuinely differ in singularities, smoothness, and finer cohomology off those subsets.