As the define goes:
A subgroup $N$ of a group $G$ is called a normal subgroup if it is invariant under conjugation; that is, for each element $n$ in $N$ and each $g$ in $G$, the element $gng^{−1}$ is still in $N$.
My Question is: Can anyone give me an intuitive explanation or an example of this concept? Why it is very important in algebra?
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.
UPDATE: example of a normal subgroup.
Let's take the same cube. Now lets allow only rotations for 180 degrees around $x$, $y$ and $z$ axis. And any combination of such rotations. This will be group $B$. $B$ is a subgroup of the group of all rotations of cube. It is a proper subgroup of the original group (each face of our cube either remains in place or is moved to the opposite position by our rotations, so not all the elements of original group are included into this subgroup).
The definition of $B$ "does not depend of frame of reference", so I am sure this is a normal subgroup. Well, I understand that "does not depend of frame of reference" is not an accurate description, but this whole question is about intuition.
By the way, looks like group $B$ consists of only 4 elements: after any combination of described rotations only 1 of 2 faces can become the front, and only 1 of 2 faces can become the top. So "and any combination of such rotations" in my definition of group $B$ can be substituted with "and identity element $e$".
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.
I learned this approach in Basic Algebra I by Jacobson. The same approach leads to ideals in ring theory.
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.
