The eigenvalues are real. Since the product is one, one has absolute value larger than one and the other smaller than one. Since the eigenvalues are different we can find two eigenvectors. One eigenvector corresponds to a stable direction, points will march towards the fixed point, but the other direction is unstable. Hence we have a saddle point or hyperbolic fixed point.
Thes means that the eigenvalues are the same and have absolute value one.
If we can find both eigenvectors the map is boring, just the identity. If the matrix is defective we can only find one eigenvector. An example is the tangent map , which corresponds to a shear along the x-axis. This situation is not very common, though. Note that in all three cases the order in which points follow trajectories depends on the sign of the eigenvalues.