On cubes, one of the faces has a checkered face, that doesn't like to cooperate with what I am doing. Can someone shed some light on what exactly this face is, and what it's purpose is?
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Basic ingredients of a mesh are triangles (tris), quadrilaterals (quads), and polygons with n sides (n > 4, n-gons). N-gon is the least often used, and indeed Blender's own support for true n-gon is fairly recent (2.63).
The lower right face of your problematic mesh is a concave quad. Normally, proper rendering of a concave polygon requires it to be decomposed/fragmented into several convex polygons, a doable operation but potentially expensive [PDF] for realtime display. Blender doesn't do such operation, blindly drawing a face for each connected edges in each polygons in a mesh, hence the "ghost face".
To avoid such occurence, manually convert the quad to tris (select the face, then press Ctrl-T) for the upper and lower part of the quad:
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$\begingroup$ Can you explain in your example why there are checkers on the bottom left? $\endgroup$ Commented Sep 30, 2013 at 20:40
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$\begingroup$ If selected, the active face is also checkered, to differentiate it from other selected faces being highlighted. $\endgroup$– AdhiCommented Oct 1, 2013 at 1:17