If I have a white noise fed into a filter what would be the output of the filter - what would you expect to see? I know that white noise has a unit power spectral density (PSD), how these thing relate to each other?
For a linear time-invariant filter, the output of the filter would be coloured noise.
Again assuming linear time-invariant filter, the effect of the filter on the output can be further characterized in terms of its power spectrum density (PSD). Specifically, for an input $X(f)$ with power spectrum density $S_x(f)$ to a linear time-invariant filter with frequency response $H(f)$, the power-spectrum density $S_y(f)$ of the output of the filter is given by:
$$ S_y(f) = |H(f)|^2 S_x(f) $$
So, for a white noise input with unit power spectrum density $S_x(f)=1$, the corresponding PSD of the output would be $$ S_y(f) = |H(f)|^2 $$