Quadratic Residue Modulo n:
$a \in \mathbb Z_n^*$ is quadatic residue of modulo n if there exists an element $x \in \mathbb Z_n^*$ such that
$$x^2 \equiv a \mod n$$
I'm not getting the intuition behind this structure, How is it helpful in Number theory. Can anybody explain it to me.
If you're asking why we study such a structure, one reason is that it makes an appearance in cryptography. Using Legendre and Jacobi symbols for quadratic residues you can break certain cryptographic systems. So in a way, studying that structure helped in improving the system and guarding against that kind of attacks.