discrete mathematics – Struggling to understand basics of complete residue system

I’m really struggling to understand the literal arithmetic being applied to find a complete residue system of modulo $n$. Below is the definition my textbook provides along with an example.

Let $k$ and $n$ be natural numbers. A set ${a_1,a_2,…,a_k}$ is called a canonical complete residue system modulo $n$ if every integer is congruent modulo $n$ to exactly one element of the set

I’m struggling to understand how to interpret this definition. Two integers, $a$ and $b$, are “congruent modulo $n$” if they have the same remainder when divided by $n$. So the set ${a_1,a_2,…,a_k}$ would be all integers that share a quotient with $b$ divided by $n$?

After I understand the definition, this is a simple example provided by my textbook

Find three residue systems modulo $4$: the canonical complete residue system, one containing negative numbers, and one containing no two consecutive numbers

My first point of confusion is “modulo $4$“. $a{space}mod{space}n$ is the remainder of Euclidean division of $a$ by $n$. So what is meant by simply “modulo $4$“? What literal arithmetic do I perform to find a complete residue system using “modulo $4$“?