Such maps are called quasiregular. There is a highly developed theory of them. Most of the classical theory you can find in the books of Yu. Reshetnyak, Space mappings with bounded distortion, AMS, 1989, and S. Rickman, Quariserular mapings, Springer, 1993.
Probably the most fundamental fact about such maps is that they are open and discrete.
In dimension 2 their structure is quite simple: they are compositions of complex analytic functions with homeomorphisms. Also if such a map is sufficiently smooth (in any dimension) then it is a quasiconformal local homeomorphism, and if smooth and defined in the whole $R^n$ then it is a global homeomorphism. But if you relax the smoothness conditions you can have many interesting maps which are subject of study in this theory, the simplest example is $(r,\theta,z)\mapsto (r,2\theta,z)$ in cylindrical coordinates in dimension 3.
But a complete answer on your question "what can be said?" will have length of a large book.