i.e. Give a description of each of the congruence classes modulo 3.
I think I understand this but I wanted to verify. In my notes I've read "If $R$ is the congruence modulo $m$ relation on the set $\mathbb Z$ of integers then $\mathbb Z/m = \{[0], [1],...[m-1]\}$".
So for $3$ we have the classes $[0],[1],$ and $[2]$. I then described each as follows:
$$[0] = \{0 + 3k\} = \{0,3,6,9,..\}$$
$$[1] = \{1+3k\} = \{1,4,7,10,...\}$$
$$[2] = \{2+3k\} = \{2,5,8,11,...\}$$
Am I correct?
The comments suggest that the main terminology you need is congruence modulo $n$. So we have $$ a\equiv b\bmod 3 \Longleftrightarrow 3\mid a-b $$ in the ring $\Bbb Z$. This has nothing to do with negative or positive numbers. It concerns all integers, i.e, $$ -1\equiv 2\equiv 5 \bmod 3 $$ for example. In the quotient ring $R=\Bbb Z/3$ these become equalities: $$ -7=-4=-1=2=5=8 $$ in $R$, and so on. So in the quotient all these numbers are just treated as one element, namely as $[2]$ in your terminology.
Application: The equation $x^2+y^2=7919$ has no integer solution.
Proof: Assume it has an integer solution. Then it also has one in the quotient $R=\Bbb Z/4$. Here we obtain $x^2+y^2=3$, which clearly has no solution there. Contradiction.