Examples of algebraic stacks without coarse moduli space?

[A^1/G_m] is one example. You can check that any G_m invariant map from A^1 to a scheme is constant. Thus the map from [A^1/G_m] to the point is universal for maps to schemes, but is not a bijection on geometric points (since [A^1/G_m] has two geometric points).

Check out Jarod Alper's thesis to learn more.