For symmetric group conjugacy classes and irreducible representation both are parametrized by Young diagramms, so there is a kind of "good" bijection between the two sets. For general finite groups see MO discussion.

Question: What are the finite groups where some "good" bijection(s) between conjugacy classes and irreducible representations are known ?

"Good" bijection is informal "definition", nevertheless I hope example of S_n and other examples listed below, may convince that the question makes sense. I think that it far too optimistic to have one unique bijection for general group, but it seems to me that for certain classes of groups there can be some set of "good" bijections. It is tempting to ask what properties should "good" satisfy, but let it be next question.

Some examples:

1) symmetric group S_n

2) Z/2Z is naturally isomprhic to its dual, as well as $Z/2Z \oplus Z/2Z$ see e.g. MO "fantastic properties of Z/2Z"

3) Generally for abelian finite groups: among all set-theoretic bijections $G \to \hat G$, some are distinguished that they are group isomorphisms. So we have not unique, but a class of "good" bijections.

4) For GL(2,F_q) Paul Garret writes: "conjugacy classes match in an ad hoc fashion with specic representations". (See here table at page 11).

5) G. Kuperberg describes relation of the McKay correspondence and that kind of bijection for A_5 (or its central extension), see here.

6) If I understand correctly here at MO D. Jordan mentions that bijection exists for Coxeter groups. (I would be thankful for detailed reference).

7) It seems that for Drinfeld double of a finite group (and probably more generally for "modular categories") there is known some analog of natural bijection. There is such remark at page 5 of Drinfeld Doubles for Finite Subgroups of SU(2) and SU(3) Lie Groups. R. COQUEREAUX, Jean-Bernard ZUBER:

In other words, there is not only an equal number of classes and irreps in a double, there is also a canonical bijection between them.