hyperbolic 3-manifold of finite volume Is there a complete description of hyperbolic 3-manifold of finite volume ?
Or similarly a classification of finitely generated torsion free subgroups of $PSL(2,\mathbf{C})$ with finite covolume? 
Is it true that any such hyperbolic 3-manifold of finite volume can be obtained as $S^{3}-K$ where $K$ is some knot? 
 A: One can regard the geometrization theorem as a classification of hyperbolic 3-manifolds of finite volume. These are the interiors of ($\mathbb{P}^2$-)irreducible compact 3-manifolds $M$ with incompressible boundary, Euler characteristic $=0$, and atoroidal (every map $\mathbb{Z}^2 \to \pi_1(M)$ factors through a map to $\pi_1$ of a boundary component).
In principle, there are algorithms for listing hyperbolic 3-manifolds with bounded volume. Some version of this algorithm is implemented for small enough volume e.g. here: https://arxiv.org/abs/0910.5043. What such a list entails is a list of finitely many cusped manifolds, and a description of Dehn surgery coefficients that give all manifolds bounded by a fixed volume (although there's some thorny issues here about exactly how one computes or compares volumes). If you want to list hyperbolic manifolds, say by rank (the number of matrices one needs to multiply and take inverses to generate the full group), then things become much trickier (see e.g. https://arxiv.org/abs/1708.01774). Listing by the minimal number of tetrahedra in a triangulation should be more feasible. 
The third part of your question is addressed here: https://mathoverflow.net/a/277968/1345. But more fundamentally, finite volume hyperbolic 3-manifolds may have multiple ($>1$) or zero cusps, and hence could not be a knot complement. 
A: There is a description of the class of geometrically finite manifolds (which include finite volume hyperbolic manifolds) in an article of B. Bowditch, Geometrical finiteness with variable negative curvature, Duke Math. J., 77 (1995) 229-274.
There is also a result asserting that hyperbolic manifolds of finite volume (or more generally analytic manifolds with $-1 \leq K \leq 0$ and Injrad $\rightarrow 0$) are diffeomorphic to the interior of a compact manifold with boundary. See e.g. Ballmann Gromov, Schroeder,  Manifolds of nonpositive curvature, Progress in math 61, section 13.
