You may already know this, as it is just a textbook definition, but here it is nevertheless:

For each surjective function f:X-->Y there is a function g: Y--> X such that for all y (which are elements of Y), f(g(y))=y.

With the aid of the axiom of choice, one can show that,

given two sets x and y, one of the following two statements holds:

(i) There is an injective function f:x-->y

(ii) there is an injective function f: y-->x

In other words, two sets can always be compared "in size," though this is not a simple matter.