Personal favorite:  The Jacobian J of a smooth curve, which is the coarse moduli space for degree zero line bundles on that curve.  If you choose a point in your curve, you can also realize the Jacobian as the stack which classifies pairs (L,t) where L is a degree zero line bundle and t is a point in the fiber of L over your chosen point (i.e. a trivialization). There's an obvious surjective map from this stack to the stack M of degree zero line bundles; just forget the trivialization.  Thus, the Jacobian is both a coarse moduli space for M and an atlas for M.