Of course the equivalence of categories mentioned by Eric A. Bunch is true only for commutative $C^\ast$-algebras. There however is a quite similar result for a wider category of non-commutative $C^\ast$-algebras: a continuous-trace $C^\ast$-algebra $A$ with Hausdorff spectrum $X$ is isomorphic to the $C^\ast$-algebra $\Gamma_0(X,\mathcal{A})$ of continuous sections vanishing at $\infty$ of some continuous $C^\ast$-bundle $\mathcal{A}\to X$; the latter is actually a Dixmier-Douady bundle, in the sense that it has typical fiber the algebra $\mathbb{K}(H)$ of compact operators on some separable Hilbert space. This construction yields to an equivalence between the category of continuous-trace $C^\ast$-algebras with Hausdorff spectrum and pairs $(X,\mathcal{A})$.
When it comes to study $K$-theory of such $C^\ast$-algebras, the above equivalence may be very useful, for if $A$ corresponds to the pair $(X,\mathcal{A})$, then $K_\ast(A)$ is isomorphic to the twisted $K$-theory ${}^{\mathcal{A}}K^\ast(X)$, which in the finite-dimensional case may be interpreted in terms of geometric objects (twisted vector bundles). For instance, let $G$ be a compact Hausdorff group. Then the dual space of $G$ is homeomorphic to the the spectrum $X$ of the $C^\ast$-algebra $C^\ast(G)$ (which is actually continuous-trace). Now, for $k\in \mathbb{N}$, let $X_k$ be the (open) subspace of $X$ consisting of (equivalence classes of) irreducible representations of $G$ of rank $k$. Then, for each $k$, there is an Azumaya bundle $\mathcal{A}_k\to X_k$ (i.e. a Dixmier-Douady bundle of finite dimension), and there is an isomorphism $K_\ast(C^\ast(G))\cong \bigoplus_k {}^{\mathcal{A}_k} K^\ast(X_k)$ (cf. http://front.math.ucdavis.edu/0201.5207 for a generalization of this example).
Of course the equivalence of categories mentioned by Eric A. Bunch is true only for commutative $C^\ast$-algebras. There however is a quite similar result for a wider category of non-commutative $C^\ast$-algebras: a continuous-trace $C^\ast$-algebra $A$ with Hausdorff spectrum $X$ is isomorphic to the $C^\ast$-algebra $\Gamma_0(X,\mathcal{A})$ of continuous sections vanishing at $\infty$ of some continuous $C^\ast$-bundle $\mathcal{A}\to X$; the latter is actually a Dixmier-Douady bundle, in the sense that it has typical fiber the algebra $\mathbb{K}(H)$ of compact operators on some separable Hilbert space. This construction yields to an equivalence between the category of continuous-trace $C^\ast$-algebras with Hausdorff spectrum and pairs $(X,\mathcal{A})$.
When it comes to study $K$-theory of such $C^\ast$-algebras, the above equivalence may be very useful, for if $A$ corresponds to the pair $(X,\mathcal{A})$, then $K_\ast(A)$ is isomorphic to the twisted $K$-theory ${}^{\mathcal{A}}K^\ast(X)$, which in the finite-dimensional case may be interpreted in terms of geometric objects (twisted vector bundles). For instance, let $G$ be a compact Hausdorff group. Then the dual space of $G$ is homeomorphic to the the spectrum $X$ of the $C^\ast$-algebra $C^\ast(G)$ (which is actually continuous-trace). Now, for $k\in \mathbb{N}$, let $X_k$ be the (open) subspace of $X$ consisting of (equivalence classes of) irreducible representations of $G$ of rank $k$. Then, for each $k$, there is an Azumaya bundle $\mathcal{A}_k\to X_k$ (i.e. a Dixmier-Douady bundle of finite dimension), and there is an isomorphism $K_\ast(C^\ast(G))\cong \bigoplus_k {}^{\mathcal{A}_k} K^\ast(X_k)$ (cf. http://front.math.ucdavis.edu/0201.5207 for a generalization of this example).