As defined in Wikipedia, the cardinalities of sets can be compared by finding a mapping between the sets.
In particular, for nonempty sets $A$ and $B$, $|A| \le |B|$ if and only if there is an injection from $A$ to $B$.
Given only the definition, is it equivalent to saying that $|A| \le |B|$ if and only if there is a surjection from $B$ to $A$?
I guess it is true because every injection has a left-inverse which is a surjection, and every surjection has a right-inverse that is an injection.
This statement is known as the Partition Principle and a consequence of the Axiom of Choice.
Interestingly, it is still unknown if the Partition Principle is actually equivalent to the Axiom of Choice (many set theorists don't think so, though).