I am aware of isometric embedding of NLS(Normed Linear Space) $X$ into $X^{**}$. I am just curious to know whether there any classification or any other kind of beautiful theorem which says that they are isometrically isomorphic.
For $\mathbb{C}^{n**}\cong\mathbb{C}^n$. I know they are isometrically isomorphic.
Any kind of assumption on $X$ is fine like Hilbert, Normal or any other.
[Upgrading from comment to answer as suggested.]
You are describing what is known as a reflexive space. See the example section of the link for a number of conditions that guarantee reflexivity. Particularly important examples of reflexive spaces include uniformly convex spaces, which include Hilbert spaces and $L^p$ spaces for $1<p<\infty$. Finite dimensional normed spaces are also always reflexive, since the dual (and therefore also the bidual) has the same dimension as the original space.
Remark
Uniformly convex spaces (or those that have an equivalent uniformly convex norm) are not the only examples of reflexive spaces, as the result of M. M. Day cited in this answer shows.