Q:

What does 1 modulo p mean?

For example, from Gallian's text:
Sylow Test for Nonsimplicity 
Let $n$ be a positive integer that is not prime, and let $p$ be a prime divisor of $n$. If 1 is the only divisor of $n$ that is equal to 1 modulo p, then there does not exist a simple group of order $n$.
I of course understand what some a mod b is (ex. 19 mod 10 = 9); but wouldn't 1 mod anything positive be 1 [edit: changed from 0]? 

A:

To say that a number $a$ is $1$ modulo $p$ means that $p$ divides $a - 1$.  So, in particular, the numbers $1, p + 1, 2p + 1, \ldots$ are all equal to $1$ modulo $p$.
As you're studying group theory, another way to put it is that $a = b$ modulo $p$ if and only if $\pi(a) = \pi(b)$ where $\pi\colon\mathbb Z \to \mathbb Z/p$ is the factor homomorphism.

