I'll interpret your question to be regarding closed subgroups of $SL(2,\mathbb{C})$, since you mention in the first paragraph closed subgroups of $O(3)$. Otherwise very little is known (see e.g. Calegari-Dunfield for non-discrete examples).
Moreover, usually people work with classification of closed subgroups of $PSL(2,\mathbb{C})$. Then the $SL(2,\mathbb{C})$ case follows by taking a central extension of the discrete subgroup of $PSL(2,\mathbb{C})$ (or a lift avoiding $-I$ if it exists).
As you mention, the compact subgroups of $SL(2,\mathbb{C})$ are conjugate into $SU(2)$. For closed subgroups of positive dimension, the identity component is a Lie subgroup, such as $SU(2)$, $SL(2,\mathbb{R})$, and some solvable (upper triangular) examples. These may be classified by considering the fixed points or invariant subsets of the action on $\mathbb{CP}^1$. I believe that this classification is fairly straightforward (if tedious), but I haven't considered the case-by-case analysis (it's possible that it exists in the literature, but I haven't done a search added: see the reference to Greenberg given by Misha in the comments below).
Otherwise, one has the discrete subgroups of $PSL(2,\mathbb{C})$. Here there is a big difference in what is known between the finitely-generated and infinitely generated cases.
The finitely-generated case (as noted by Yves de Cornulier in the comments) is the classification of Kleinian groups (historically stronger assumptions were made on such groups, and some do not require them to be finitely generated). In a strong sense, this classification was carried out culminating in the Ending Lamination Theorem. There are some caveats here though: the proof has not appeared in the case of decomposable (free product) groups (or more generally groups amalgamated over finite subgroups). Moreover, the classification gives these groups indirectly. It states that the group is the fundamental group of a compact hyperbolic orbifold (possibly with boundary) together with some ending data, which is a set of conformal structures and ending laminations associated to subsets of the boundary. Once one is given such data (which can be encoded combinatorially together with a weighted train track and a point in an appropriate moduli space), it is still a non-trivial step to actually compute the group (e.g. in terms of a set of matrix generators). So this theorem is really giving just a 1-1 correspondence between conjugacy classes of Kleinian groups and some topological and conformal data.
As an example, the 2-parabolic case has a well-understood classification. In this case, the free product forms the
"Riley slice" of Schottky space (the rainbow shaded region in the picture).
The interior contains many more 2-parabolic generated discrete groups which are
not free products, such as 2-bridge knot groups.
In principle, these all have a classification (although there
are still some issues regarding the complete classification of
the parabolic generating sets).
For the non-finitely-generated case, there is not even
a conjectured classification. Tommaso Cremaschi is beginning
to consider examples (although there were previous examples known
to experts).
