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][1] 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][2] 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][3].


  [1]: http://en.wikipedia.org/wiki/Vanish_at_infinity
  [2]: http://en.wikipedia.org/wiki/Multiplier_algebra
  [3]: http://en.wikipedia.org/wiki/Spectral_triple