Definition 19.4:

The universal covering is a covering $e:(E, e_0)\longrightarrow (X, x_0)$ such that for every covering $p:(Y, y_0)\longrightarrow (X, x_0)$ there is a unique homomorphism $\psi:(E, e_0)\longrightarrow (Y, y_0)$, that is a continuous surjection $\psi$ such that $p\circ \psi = e$.

The universal covering if it exists is unique and one can establish the existence of a universal covering for a reasonable nice class of topological spaces $X$.