What exactly are equivalence relations and equivalence classes? The latter is giving me the most trouble; I've tried to read multiple sources online but it just keeps going over my head.
Example question: What are the equivalence classes of 0 and 1 for congruence modulo 4?
On
take any number $n$ and divide it by $4$ and consider the reminder the values of remainder range from $0$ to $3$
as a result there are 4 classes :-
equivalence class of $0$ is $\{4,8,12,........\}$ (multiples of 4).
equivalence class of $1$ is $\{5,9,13,........\}$ numbers of the form $4k+1$
On
At the beginning you have a large set of "things" that differ in millions of ways, say the set of all people living on earth at the moment. But you are mainly interested in languages. Not knowing yet how many languages there are in the world you nevertheless are able to group all these people according to their native tongues: Call two people equivalent if their native tongues coincide. This simple device puts some structure onto the cloudy multitude of people: They now fall into groups having the same native tongue, and that exactly is the topic that interests you. Each of these groups is called an equivalence class with respect to the relation "are of same native tongue". Note that you were not required to have a list of all existing languages at the start; but now you can count the number of languages, and you can also form a "set of representatives" by selecting a member of each language group.
Heuristically: Let us take a finite set $S= \{s_1,...,s_n\}$. Now we "glue" together some elements, for example we could glue together $s_1$ and $s_2$, if we consider them to be "same object" we obtain a new set $S^*= \{s_1=s_2, s_3,...,s_n\}$. How should we call this first object in $S^*$? We can call it by $s_1$ or $s_2$ because it is determined by one of the two, but it is clear that it is not the same object as $s_1$ or $s_2$ in $S$. What about calling it equivalence class of $s_1$ and $s_2$? Similarly we can call equivalence relation the glueing operation we perform on the set $S$.
Formally: This time the set $S$ is not finite anymore, simply because we don't need it. Let us cite the formal definition of equivalence relation from Wikipedia.
To translate what we said above now we glue the elements $s_1$ and $s_2$ of $S$ if and only if $s_1\sim s_2$. Why do we need this complicated definition? Because we can now make a partition of the set $S$, let us call $[s]=\{t\in S\; | \;t\sim s\}\subset S$ the set of all the elements in $S$ which are in relation with $s$, and make this construction for every element of the set.
It is clear that can happen $[s] \cap [t] \neq \emptyset$ for some elements $s,t \in S$, but using the definition of equivalence relation above you can easily verify that this is indeed a partition of the set $S$. Where this essentially means that if $[s]\cap [t] \neq \emptyset$ then $[s]=[t]$.
Now the last step is simply to give a name to the subset $[s]\subset S$, how do you like equivalence class of $s$?
Example: Let us examine your proposed example. Take $S=\mathbb{Z}$, the set of integers number. and the equivalence relation to be $a \sim b \Leftrightarrow a = 4k +b$ for some $k\in \mathbb{Z}$. Then you can use the machinery above for $s=0$: $$[0]= \{n \in \mathbb{Z} \:|\: n= 4k \text{ for some } k \in \mathbb{Z}\}= \{...,-12,-8,-4,0,4,8,12,...\}.$$ And for $s=1$: $$[1]= \{n \in \mathbb{Z} \:|\: n= 4k +1 \text{ for some } k \in \mathbb{Z}\}= \{...,-13,-9,-5,-1,3,7,11,...\}.$$