Characterize the Equivalence Classes of a Language
The notion of equivalence is that of indistinguishability. Given a language $L \subseteq \Sigma^{\ast}$, we say that strings $x$ and $y$ are indistinguishable wrt. $L$, written $x \, I_L \, y$, if we have that for every $z \in \Sigma^{\ast}$ that $xz \in L \iff yz \in L$.

The Myhill-Nerode theorem tells us that a language $L$ is a regular language iff the set of equivalence classes of $I_L$ is finite.

To show that $L = \{ w \in \{ a,b\}^{\ast} \mid \exists n \geq 0. w = a^n b^n \}$ is not regular, we simply have to show that the set of equivalence classes of $I_L$ is infinite.

So consider a string $w \in \{ a,b\}^{\ast}$. The set of all strings $w$ not of the form $a^i b^j$ where $i \geq j$ forms an equivalence class, since for no $z$ we have that $w_1 z \in L$ if $w$ is not of the form $a^i b^j$.

Moreover, every string of the form $a^i b^j$ where $i \geq j$ can only be indistinguishable from itself. So every such string forms an equivalence class containing only this single string.

Consequently, $I_L$ has infinitely many equivalence classes.