a dominant morphism
When you map one variety into another, the image might fill up the target densely or it might land inside a thin slice. A dominant morphism is one whose image is dense — it spreads out across essentially all of the target rather than collapsing into a smaller subvariety. The projection of the plane onto a line, sending (x, y) to x, is dominant; the constant map sending everything to a single point is as far from dominant as possible.
Let f: X -> Y be a morphism of irreducible varieties. It is dominant if its image f(X) is dense in Y in the Zariski topology, equivalently if the closure of the image is all of Y. The algebraic translation is clean and useful: f is dominant exactly when the induced k-algebra homomorphism on coordinate rings f*: k[Y] -> k[X] is injective, which in turn extends to an inclusion of function fields k(Y) -> k(X). So a dominant morphism is precisely one that does not kill any nonzero function on the target. This makes function fields functorial in the right way and is the reason dominant maps, not arbitrary maps, are the morphisms relevant to birational geometry; a birational map is a dominant map inducing an isomorphism of function fields. For a dominant morphism of irreducible varieties one has dim X is at least dim Y, and the generic fiber has dimension dim X minus dim Y.
Dominance is the condition that lets you compare varieties through their function fields and underlies the entire birational viewpoint, field extensions k(Y) inside k(X) encoding the geometry of the map. Two honest caveats. Dominant is about density of the image, not surjectivity: the image of a dominant morphism need not be all of Y, only dense, and it always contains a dense open set (Chevalley) but can miss a thin closed piece — for example the map A^1 -> A^1 by t -> t is trivially onto, but A^2 -> A^2 by (x, y) -> (x, xy) is dominant yet misses the points with x = 0, y not 0. And the fiber-dimension statement is about the generic fiber; special fibers can jump in dimension, as that same example shows over x = 0.
The map f: A^2 -> A^2 by (x, y) -> (x, xy) is dominant: f* sends the coordinate functions to x and xy, an injection of k[u, v] into k[x, y], so the image is dense. Yet f is not surjective — its image misses every point (0, c) with c not 0, since x = 0 forces xy = 0. Density without surjectivity.
Dominant means dense image, which is strictly weaker than surjective; the image always contains a dense open set but may miss a thin closed piece.
Dominant is density of the image, not surjectivity. By Chevalley the image is constructible and contains a dense open set, but it can fail to be all of Y; and dim X >= dim Y is the generic-fiber statement, with special fibers allowed to jump.