Keel-Mori's theorem says an algebraic stack with a finite diagonal over a scheme S has a coarse moduli space. What is an example of an algebraic stack without coarse moduli space?
[A^1/Gm] is one example. You can check that any Gm invariant map from A^1 to a scheme is constant. Thus the map from [A^1/Gm] to the point is universal for maps to schemes, but is not a bijection on geometric points (since [A^1/Gm] has two geometric points).
Check out Jarod Alper's thesis to learn more.