From Wolfram
The residue classes of a function $f(x)$ mod $n$ are all possible values of the residue $f(x)~(\pmod n)$. For example, the residue classes of $x^2~(\pmod 6)$ are $\{0,1,3,4\}$.
From Wikipedia
Each residue class modulo $n$ may be represented by any one of its members, although we usually represent each residue class by the smallest nonnegative integer which belongs to that class (since this is the proper remainder which results from division). Any two members of different residue classes modulo $n$ are incongruent modulo $n$. Furthermore, every integer belongs to one and only one residue class modulo $n$.
So, the Wolfram means that the residue class of $f(x)=x~ (\pmod n)$ is $\{0,1,2,...,n-1\}$ while Wikipedia says that residue classes are $[0],[1],\dots,[n-1]$ where $[x]=\{x+kn\}$ for integer $k$ and that is contradicting.
Also, the Wikipedia definition is affirmed by this post on StackExchange.