Can someone explain the definition of four square roots as it pertains to groups in Z*p?
"$x$ has four square roots" means there are four distinct elements $r$ in the group such that $r^2 = x$.

For example, in $\mathbf{Z}_{15}^\times$ the element $1$ has four square roots, which are $1$, $4$, $11$, and $14$ (note that $11 \equiv -4$ and $14 \equiv -1  \pmod{15}$).