The matrices you are looking for are exactly those that have spectral radius (the max. of the absolute value of the eigenvalues) strictly less than one.
I do not know whether there is a more specific name.
(A matrix such that a finite power would be exactly the zero-matrix would be called nilpotent; but this is a different property.)
Regarding the invertibility of $I-A$.
Note that (first only formally) $(I-A) (I + A + A^2 + \dots )=I$
To make this rigorous it suffices to show that $(I + A + A^2 + \dots )$ converges.
This can be done by noting that the spectral radius is 'almost' a matrix norm;
more precisely, for $\varepsilon>0$ and all sufficiently large $k$ one has $||A^k|| \le (r + \varepsilon)^k$ where $r$ is the spectral radius. Now, you just have to sum a geometric series. For some more details and or background see e.g. http://en.wikipedia.org/wiki/Spectral_radius and http://en.wikipedia.org/wiki/Matrix_norm