A very good reference for this is the recent book of De Loera, Rambau and Stantos called "Triangulations: Structures for Algorithms and Applications". They have a whole chapter on regular triangulations. I don't know if the book is in print yet. It's possible that careful googling could reveal drafts of the book that are still online.
The classic example of a triangulation of a point set that is not regular is the following: Draw two concentric triangles with vertices {1,2,6} and {3,4,5}, the outer triangle has vertices {1,2,6} and the inner triangle has vertices {3,4,5}. The following is a triangulation: {16,26,21, 34,45,43, 36,56, 13,14, 24,25}, where {ij} means draw an edge from i to j.
You can see that this is not regular by assuming that {3,4,5} were not lifted above the plane, and then trying to lift vertices, 1,2,6 to get the remaining faces. You will find that you need the heights to satisfy height(1) < height(2) < height(6) < height(1). This is spelled out in complete detail in Sturmfels' book.