I just wanted to add two more examples about torsion in cohomology groups of low degree that came into my mind reading the above (great) answers:
- Any torsion element in $H^2(M, \mathbb{Z})$ for a space $M$ can be realized as the first Chern class of a complex flat line bundle.
- Similar to this, you may know that elements in $H^3(M, \mathbb{Z})$ correspond (up to some equivalence) to twists in twisted K-theory. Now, if that class is torsion, you get a very nice description of twisted K-theory via modules over bundle gerbes. Or, if you don't like twisted K-theory, the torsion elements in $H^3(M,\mathbb{Z})$ correspond to (stable equivalence classes) of those bundle gerbes, which allow a (finite dimensional) bundle gerbe module.
