Following up on Charles' comment to Kevin's answer, torsion can be helpful in determining whether or not a manifold is orientable: $H_{n-1} (M; Z)$ is torsion-free when M is orientable and has torsion subgroup Z/2 when M is non-orientable. For surfaces, this means orientability can be detected from H_1, which is quite nice.

On the other hand, you don't really need to pay attention to torsion to see the difference between orientability and non-orientability. A closed (connected) n-manifold M is orientable iff $H_n (M; Z) = Z$, and non-orientable iff $H_n (M; Z) = 0$. The same statements hold with integral coefficients replaced by real coeffients.

This is all in Hatcher's section on Poincare Duality.