What is $ \mathbb{N}^2 / R $? I understand, that R is a symbol for an equivalence relation and $\mathbb{N}$ are the natural numbers.
But what is meant with  $\mathbb{N}^2 / R $ ?
 A: This is the set of all equivalence classes of $R$.
A: An equivalence relation $R$ on a set $S$ tells us whether two elements $a,b$ are related or not.


*

*in simple language, $a$ and $b$ are friends
So when considering an integer $a$, we can define $\bar a=\{x\in S\mid xRa\}$ of all elements that are in relation with $a$.


*

*$\bar a=friends(a)$
Note that when $a$ and $b$ are in relation then $\bar a=\bar b$ 


*

*the friend of my friend is my friend too
In the end, we can split $S$ into classes of related elements. 


*

*$S$ is split into clans, all people are friends into one clan
The quotient $S/R$ is the set of all equivalence classes of $R$


*

*we are not interested in people anymore, just $clan_1,clan_2,clan_3,...$

Let's take an example in $\mathbb N$
We define $a\,R\,b\iff a\text{ and }b\text{ have same parity}$
We see there are $2$ classes, the even numbers and the odd numbers.
They can be represented by $\bar 0=\{\text{even numbers}\}$ and $\bar 1=\{\text{odd numbers}\}$, but we could have chosen as well $\bar 2$ and $\bar 7$, or $\bar 8$ and $\bar 1$, it does not matter which representative  we choose, they are the same classes of even and odd numbers.
Now $\mathbb N/R=\{\bar 0,\bar 1\}=\{\text{even numbers},\text{odd numbers}\}$
Note that $\mathbb N/R$ has only $2$ elements (classes), but their nature is different from integers, they are themselves sets of integers.

Let's take another example in $\mathbb N$
We define $a\,R\,b\iff |a-b| \text{ is divisible by 7 }$
This time there are $7$ classes, for sake of simplicity we will note them $\bar 0,\bar 1,...,\bar 6$
Now $\mathbb N/R$ is the set of all classes of numbers modulo $7$
$\bar r=\{x\in\mathbb N\mid x\equiv r\pmod 7\}$

In your question, you asked for $\mathbb N^2/R$, so there is some relation between points in the plane with positive integer coordinates.
$\mathbb N^2/R$ is the set of classes of points that share the same property.



*

*For instance, such a relation $R$ could be: "the points belong to the same vertical line"
This means that we do not care about the second coordinate, thus $(n,a)$ and $(n,b)$ belong to the same class.
We could as well consider only the classes $(n,\square)$ or simply $\bar{n}$.
In this case, $\mathbb N^2/R$ can be identified to $\mathbb N$.



*

*For instance, such a relation $R$ could be: "the points belong to the same line issued from origin"
These lines all have a positive slope, which since it passes through points $(a,b)$ with integer coordinates, is a rational number.
In this case, $\mathbb N^2/R$ can be identified to $\mathbb Q^+$.
A: A very important equivalence relation on $\mathbb{N}^2$ is 
$$(a,b)\simeq (c,d) \iff a+d = b+c$$
( one thinks of it in terms of $a-b = c-d$, but the difference might not lie in $\mathbb{N}$, so we stay inside $\mathbb{N}$.
The equivalence classes can be indentified with $\mathbb{Z}$. 
