Are you looking for a complete classification? It seems unlikely that it can be achieved.
At any rate, talking about finitely generated, residually finite groups the first result that comes to my mind (apart the well-known fact that every free-group is residually finite) is the following powerful result:
Theorem (Malcev, 1940) Every finitely-generated linear group is residually finite.
See
A. I. Malcev: On the faithful representation of infinite groups by matrices, Mat. Sb. 8 (50) (1940), 405-422; English transl., Amer. Math. Soc. Transl. (2) 45 (1965), 1-18. MR 2, 216.