An elementary example everyone should know is
Here * is a point and G is an algebraic group. Its sections on a scheme X form the groupoid of principal G-bundles on X. Because principal G bundles are locally trivial, the coarse moduli space is a point.
If M is a stack, it determines a presheaf \pi_0(M), whose sections are the isomorphisms classes of objects of M. One way to think about the coarse moduli space is as a space representing the sheafification of this presheaf in whatever topology you are working with.
Starting from this example, you should also be able to work out the issues surrounding [X/G]. In fact, it's good to revisit BG after working out the general case. But I won't do it here since you asked for one example only. In fact, maybe I should have said
I learned about stacks in the days before all these nice books, so I'm not sure about a reference for anything I write about them. But the statements here should all be pretty clear from the definitions.