Sofie Verbeek's answer is perfect, but let me go back to the definition of a universal covering space. There are two different approaches: Some (perhaps most) authors define a universal covering $p : E \to X$ of a connected space $X$ to be one such that $E$ is simply connected, other authors define it by the property that it covers any covering $p' : E' \to X$ with a connected $E'$.
In my opinion the second definition is the better one because it is explains the name. If you accept this point of view, then it is a theorem that a simply connected covering of a connected and locally path connected $X$ is a universal covering.
A nice reference is Chapter 2 Section 5 of
Spanier, Edwin H. Algebraic topology. Vol. 55. No. 1. Springer Science & Business Media, 1989.
It contains a number of interesting results and examples (e.g. of a non-simply connected universal covering space of a of a connected and locally path connected space).