This machinery is somewhat high powered, but it should explain what the book is really trying to get at.
Hall subgroups are generalizations of Sylow subgroups for multiple primes. If we denote by $\nu_G(p)$ the largest power of $p$ dividing $|G|$, we see that Sylow $p$-subgroups are those subgroups of order $\nu_G(p)$. Hall $\pi$-subgroups, where $\pi$ is a set of prime numbers, are subgroups of order $\prod_{p\in \Pi} \nu_G(p)$. Alternatively, a Hall $\pi$-subgroup $H$ is a subgroup of order divisible by each $p\in \Pi$ such that $[G:H]$ is coprime to $|H|$. Note that Hall subgroups of a group $G$ may not always exist for each $\pi\subset\{p\in \mathbb{P}|p\mid |G|\}$ - actually, this holds if and only if $G$ is solvable.
As it turns out, a lot of the stuff that works for Sylow subgroups works for Hall subgroups. In particular, when Hall subgroups exist, they are all conjugate, so much like how normal Sylow subgroups must be unique, so must normal Hall subgroups. This is part of Hall's theorem, which I will not prove here (but it follows fairly easily by inducting on the size of a minimal normal subgroup).