in a simple plain, when you move a vertex in a way that it can't be in a same height with other edges, blender creates an edge between edges to keep your shape as it is.
Take a look at this:
There is couple of ways to fix it, First you can add a little bit
bevel to the mesh and then a
subdivision to give blender more edges to play with(easier to do but heavier mesh).
Or you can add
triangulate modifier and do some "edge rotate" on edges that you want to be in other direction. (lighter mesh but harder to do)
Try removing doubles by selecting all and pressing w then remove doubles.
Also,you have ngons which is automatically converted into triangles when rendering,so if it is not flat it will appear like this,