gn.general topology - Which is the correct ring of functions for a topological space?

I thought I'd record the misgivings in my comment above as an actual "answer". I think that one needs to be clearer about just what class of topological spaces one is considering. As I said above, even if one starts by restricting attention to Hausdorff spaces, the LCHff class is atypically nice (and the CHff case extraordinarily atypically nice, as passing categorists might attest). Saying that function spaces are dual to topological spaces is a great maxim, but like all maxims needs to be wielded with a modicum of care and not always as broadly as salespeople would have you do...

In this kind of broad generality, my first instinct would be to see what Gillman and Jerison's book "Rings of continuous functions" has to say.

As several people have said above, when X is not compact then one needs to decide how interested one is at "behaviour at infinity", and choose the most appropriate algebra of functions to reflect this. If you really don't care much, then $C_0(X)$ seems natural although as I said above this might only be good for LCHff X, and there are plenty of quite interesting topological spaces which are not locally compact...

I rather suspect that when $X$ is Hausdorff, $sigma$-compact but non-compact, then $C(X)$ should have a natural Frechet algebra structure, and there might be some work done on this class of examples.

By the way, if $X$ is a metric space then there is a case that one should be looking at the algebra of Lipschitz functions on X (Nik Weaver has championed this viewpoint in the past). However, since this is not a $C^*$-algebra, it might not meet the standards of Proper Functional Analysis and should perhaps be consigned to the dustbin of history (or not, depending on your point of view).