I am currently studying group theory and I realized that most concepts we study are just definitions on which we build theory. I do understand that some theorems are beautiful and don't need any practical use to be interesting to study.
My question now is: What are the motivations behind the following concepts:
Normal subgroups
Permutation groups
Right and Left cosets
On
Permutation groups.
Theorem. (Cayley) Let $G$ be a group, then $G$ is isomorphic to a subgroup of $S(G)$.
Proof. Let $g\in G$ and let define the following element of $S(G)$: $$\sigma_g:\left\{\begin{array}{ccc}G &\rightarrow&G\\h&\mapsto&gh\end{array}\right.$$ The inverse of $\sigma_g$ is $\sigma_{g^{-1}}$ and more generally $\sigma_{gg'}=\sigma_g\circ\sigma_{g'}$. Let define the following group morphism: $$\varphi:\left\{\begin{array}{ccc}G&\rightarrow&S(G)\\g&\mapsto&\sigma_g\end{array}\right..$$ $\varphi$ is injective, therefore $\varphi$ is an isomorphism between $G$ and a subgroup of $S(G)$, namely $\textrm{im}(\varphi)$.$\Box$
Remark. In short, $G$ acts on itself by left translation and this action is faithful (it is even a free action).
Remark. In a way, this theorem tells us that it is sufficient to understand permutation groups.
Remark. When $G$ is finite, this theorem tells us that $G$ is a subgroup of $S_n$ for a given $n$.
Any answer to this question is going to be lacking because you could write a book on these concepts and why they are important. So I'll just make some brief mentions of why these are important from a theoretical standpoint (there were some excellent mentions of historical significance in the comments).
The concept of normal subgroups and cosets are linked. One reason for studying them is that it is a way to created new groups from old. Also you can learn about the group from studying them.
If $H$ is a normal subgroup of $G$, then the left (or right) cosets of $G$ form a group. The multiplication is $(aH)(bH) = (ab)H$. This is a big reason why normal subgroups are important.
It is also true that normal subgroups arise as kernels of group homomorphisms. If $\varphi: G \to G'$ is a group homomorphism, then $\ker(\varphi)$ is a normal subgroup of $G$. Even further, every normal subgroup of $G$ is the kernel of some homomorphism from $G$ to another group.
Permutation groups are a fundmental mathematical object. Consider a bijective function from a set $X$ to itself. What does this function do? It permutes the elements of $X$.
This is just one example of permutations arising very naturally in mathematics. Other places where permutation group arise in very beautiful and surprising ways is the theory of covering spaces in Algebraic Topology, group actions (which are an excellent tool for studying groups), Combinatorics (permutations are everywhere here), and many more.