The recent paper On the Erdos distinct distance problem in the plane Authors: Larry Guth, Nets Hawk Katz prodded me to get a non-trivial example. Here is what I cannot find: an example of 12 distinct points in the plane with only 5 different distances between points. The regular 12-polygon has 6 different lengths but I cannot do better. http://oeis.org/A186704 implies that there is one>
Here goes my poor explanation:
Take a regular hexagonal lattice with distance 1 between nearest neighbors, and choose a 15-point equilateral triangle in this lattice (15 is a triangular number). Remove the 3 vertices of the triangle. You'll be left with 12 points and 5 distinct distances.
Edit: Just checked the OEIS reference, and it's available on Google Books. The picture you want is on page 200.
If I may supplement Logan Maingi's apposite answer with a snapshot of the page to which he refers:
(I couldn't resist including the surrounding conjecture.)