If you want to trade spaces for algebras of functions, which is the basis of noncommutative topology, then take the following algebras for the following spaces:
- If $X$ is compact Hausdorff, you want $C(X)$.
- If $X$ is locally compact Hausdorff, but not compact, you want $C_0(X)$. Then $C(X)$ corresponds to the one point compactification of $X$.
- If $X$ is compact, but not Hausdorff, then $C(X)$ corresponds to some type of "Hausdorffization" of $X$.
Actually $C(X)$ and $C_0(X)$ are the same if $X$ is compact, but you want to denote it $C(X)$ to emphasize the fact that the algebra is already unital. Otherwise, when you add a unit, you take the one point compactification of a compact space which adds an extra point, which is not what you want.
You are correct in that if you want the Stone-Cech compactification, you take $C_b(X)$. This can also be obtained by forming the multiplier algebra of $C(X)$ (if $X$ is compact Hausdorff).
Now let's suppose you have some additional structure, like $X$ is a compact manifold. Then you probably want the $C^\infty$-functions on $X$. However, these can be recovered from $C(X)$ as those operators whose iterated commutator with the Dirac operator is bounded. This inspired the notion of a spectral triple.