the significance of different contours lies in whether or not your contour encircles (if its a closed contour) the singular points of g(z)
singular points are just where g(z) is undefined, so here, where the denomenator equals 0
where z=1 or -1
so a contour that doesnt encircle these points, would be
integrating g(z) around this contour gives 0
this is by the residue theorem, closed contour integration theorem... something along those lines
for a contour that encircles the singularities
e.g.
you have to calculate the residues of g(z) at the points of singularities