The document I am reading says that if $\pi:S^3\longrightarrow L(p,q)$ is a covering map satisfying $\pi(x)=\pi(-x)$, where $p$ is even, then $L(2,1)$ as an intermediate cover and $L(2,1)$ is $RP^3$.
So I have two questions: What is the universal covering map for Lens spaces $L(p,q)$? And what does intermediate cover mean?
It is a theorem that the composite of two covers is again a cover, as long as the "final" map has finite fibers. Precisely, if $p: X \to Y$ and $q: Y \to Z$ are covers and $q^{-1}(z)$ is finite for each $z \in Z$, then the composite $q \circ p: X \to Z$ is again a covering map. Hence $Y$ is a cover intermediate to the one $X \to Z$.
Finally, your Lens spaces come from coverings $S^3 \to L(p,q)$ (via the quotient by the free action). This is a cover and $S^3$ is simply connected, so $S^3$ is already your universal cover.