Gelfand duality asserts that $C(-)$ is an anti-equivalence from the category of compact hausdorff spaces to the category of commutative unital $C^{\ast}$-algebras. Now, for a continuous map $f : X \to Y$ it is not hard to show that
$f$ is surjective $\Longleftrightarrow$ $C(f) : C(Y) \to C(X)$ is injective
Sketch of proof: $\Rightarrow$ is trivial, and $\Leftarrow$ follows from Tietze extension theorem $\square$. By the way, we also have:
$f$ is injective $\Longleftrightarrow$ $C(f) : C(Y) \to C(X)$ is surjective
Therefore, $C^{\ast}$-subalgebras of $C(X)$ (i.e. closed unital $\ast$-subalgebras) correspond to surjective maps $X \to Y$, where $Y$ is compact hausdorff. It is well-known that these maps are quotient maps. The partial order of $C^{\ast}$-subalgebras of $C(X)$ is anti-isomorphic to the partial order of quotients of $X$.
