let $A$ is a $C^*$ algebra. and $S$ is any subset of $A$. then $C^*$ algebra generated by $S$ is intersection of all sub algebras containing $S$.
if $S=\{a\}$ then can say something about structure of $C^*(a)$
where $C^*(a)$ is $C^*$ algebra generated by $a.$
or generally for any set $S$.
also what it mean to say norm closure ??
On
Indeed, for arbitary $a \in A$ we can hardly say something about the structure of the algebra $C^*(a)$. But if $a$ is normal ($a^* a = aa^*$), then Gelfand theorem tell us $C^*(a) \cong C_0(\text{sp}_A(a) \setminus \{0\})$, where $\text{sp}_A(a) \subset \mathbb{C}$ is the specturm of $a$.
There are other results for special elements (or special relationships), for example, see Farah's Book Combinatorial set theory of C*-algebras, Lemma 2.3.8.
For arbitary subset $S \subset A$, all we know is what Aweygan said, $C^*(S)$ is the norm closure of the $*$-algebra generated by $S$, where the $*$-algebra generated by $S$ is the algebra generated by $F = \{a \in A : a \in S \text{ or } a^* \in S\}$, where the algebra generated by $F$ is all the polynomials with variables in $F$ and complex coefficients.
Depending on what you mean by "structure", this question cannot be adequately answered, as every C$^*$-algebra is realizable as a C$^*$-subalgebra of $B(H)$ for some Hilbert space $H$ (hence is the C$^*$-subalgebra of $B(H)$ generated by itself).
Given a subset $S$ of a C$^*$-algebra $A$, put $T=S\cup S^*$, where $S^*=\{a^*:a\in S\}$. Then $C^*(S)$ is the norm closure of all polynomials in elements of $T$.
Now what is the norm closure? In general a linear subspace of a Banach space (or more generally a normed space) need not be closed in the norm topology, hence need not be complete. But we want completeness, so we take the norm-closure of the subspace. This is still a subspace (or a subalgebra, or a $*$-subalgebra depending on the situation), which is easily verifiable, but now we have completeness.