Is there any intuitive understanding of normal subgroup?

Let's take a look at the group of rotations of cube. It has a subgroup of rotations around vertical axis. This subgroup (let's call it $A1$) has 4 elements: rotate the cube for 0, 90, 180 or 270 degrees.

There is another subgroup: rotation around one of horizontal axes. Let's call it $A2$.

Subgroups $A1$ and $A2$ are obviously different. But still they look so very much alike! If there was someone else looking at our cube from different angle he could even fail to understand my descriptions of $A1$ and $A2$ "correctly" and confuse $A1$ with $A2$.

This is because $A1$ and $A2$ are conjugated. The $g x g^{−1}$ actually means "look at $x$ from another point of view", and $g$ defines this "point of view".

Subgroup is normal if it is very "symmetric". No matter from which point you look at the whole group $G$ the subgroup $N$ remains at place.

Quotients of groups are only well defined if we take the quotient over a normal subgroup. Another way of writing your definition is that $N$ is normal iff $gN = Ng$, so the set of left cosets equals the set of right cosets, which makes the quotient group $G/N$ well-defined. That is, in my opinion the key reason to be interested in them. Otherwise, normal subgroups keep popping up literally everywhere in group theory. For example, when we look at solvable groups (Galois theory), they are important.

A congruence in a group $G$ is an equivalence relation $\equiv$ in $G$ that is compatible with the operation of $G$: $$ a \equiv b, \ a' \equiv b' \implies aa '\equiv bb' $$ The quotient $\overline G = G\,/\equiv$ is then a group.

It is easy to prove that, when $\equiv$ is a congruence in $G$, the equivalence class of $1$ is a normal subgroup $N$ of $G$ and the equivalence classes are the cosets of $N$.

Conversely, if $N$ is a normal subgroup of $G$, then the relation defined by $a \equiv b$ if $a^{-1}b \in N$ is a congruence relation in $G$ whose equivalence classes are the cosets of $N$.

So, normal subgroups are natural objects when you consider congruences. The other equivalent characterizations of normality, such as $aN=Na$ and invariance under conjugation, follow easily.