Although some authors require a covering map to be surjective by definition, I am, in particular, talking about the definition given by Hatcher:
A covering space of a space $X$ is a space $\tilde X$ together with a map $p:\tilde X\to X$ satisfying the following condition: There exists an open cover $\{U_\alpha\}$ of $X$ such that for each $\alpha,\ p^{-1}(U_\alpha)$ is a disjoint union of open sets in $\tilde X$, each of which is mapped by $p$ homeomorphically onto $U_\alpha$. We do not require $p^{-1}(U_\alpha)$ to be nonempty, so $p$ need not be surjective.
I don't understand the remark made at the end. Since $\{U_\alpha\}$ covers $X$ and each member is the homeomorphic image of some open set in $\tilde X$, $p$ should be surjective. How come that it is possible that $p$ is not? Or did the author mean that $p$ only maps some part of $p^{-1}(U_\alpha)$ onto $U_\alpha$ whenever $p^{-1}(U_\alpha)$ is nonempty?
What is an example of a nonsurjective covering map?