Flat quads (or, why is my mesh so funky?) I think my question relates to a fundamental of 3D modeling that I haven't picked up on yet. It's very simple:

Let's say I have a flat quad. A plane on the XY plane inserted into am empty blender file is a great example. Now let's say I start moving each of the 4 vertices around on in any direction, especially the Z axis. 

At this point, there would still be one face, but its extremely unlikely that the face would be planar (I just learned this word). So blender does its best to make sense of what this should look like.

The problem I'm having is that it seems like my quads are always like this, but they never look the way I want. There are edges and seams that look like they are dangling in space, and what tends to happen is that my mesh looks great when viewed from most angles, but if you look at it from just the right angle, you notice the dangling edges or sometimes even some kind of impossible geometry. And so I might try to nudge the problematic vertices around, but this often seems to either make it worse, or just hide the bad geometry until you see it from a different angle.

For me, this always seems to happen whenever I'm trying to move vertices to add roundness to something that is originally square. 

There is another question on here about how to actually flatten the quads. That's great, and I might use some of the techniques mentioned there. My question is:


Why does this happen? (Can someone explain this better?)
What are some tips and tricks to avoid this situation?
What are some tips and tricks to mitigate this situation?


EDIT Here is an illustration:

The overall mesh as viewed from a distance looks decent (for my skill level)



But if you get closer and look from a certain angle...



Super close up:



The "edge" that is pointed to by the arrow is not actually an edge, but some kind of artifact created when blender tries to show the quad. The way the vertices are positioned, it shouldn't actually be possible to see the sliver of the face that I marked in red, since this face is supposed to be around the corner from the edges. The edges should form the silhouette on the right side of my figure. 



Here is the side view. Maybe it is the angled topology that skews the face in a funny way?

But I'm not looking for help with this one particular issue, but rather a better understanding of why this happens, and ideas to prevent this in future models.
As you already suspect, in three-dimensional digital graphics computer basically only know how to draw two things: Lines and Triangles
There are a few other primitives types (point clouds, volumes, voxels etc.) but for the sake of simplicity lets focus on the former two.
Historically speaking, in the first 3D modelling software you could only model with triangle based geometry, because it's the only unequivocally definable 3D structure with three points. Even Blender itself a few years ago as of versions pre 2.63 (when BMesh geometry engine was introduces) was still only able to define faces with 3 or 4 sides, not more.
As technology evolved it started being practical to define faces with more than three sides.
A quad, as you correctly stated is not unequivocally defined by those four points, it is not however mathematically undefined. Since computers only know how to draw triangles, in the background the software is trying to silently triangulate the shapes for you as best as it can, as you can see bellow.

Simple shapes are trivial, but more complex shapes (where supported by the software) are even more difficult to triangulate, and may often generate artifacts from unexpected or uncontrollable tessellation. That is one of the reasons why you will often hear people say you should strive to have a quad dominant topology.
Blender doesn't explicitly allow tessellation control, but other software or mesh engines some times have tools that allow controlling how a shape is triangulated (as is the case of 3DSMax 3D Polys, but not meshes).

TL;DR
Should you be worried by this? No, not in most cases.
This is normal behavior and non-planar shapes are totally expected (if not the norm) everywhere in your mesh. Modelling with planar faces everywhere would be extremely hard, or certain shapes would be downright impossible to achieve.
Some extreme cases can some times cause issues and should be avoided though, like concave planar faces, or very high distortion quads, but most others occurrences should be fine.
There are builtin tools in Blender to help with these, like the Mesh &gt; Cleanup &gt; Make Planar Faces, to help detect where issues might occur.
What you seem to be missing is the concept of subdivision. Except for hard-edge models most organic shapes are frequently modeled using a Subdivide Modifier, that will take care of automatically smoothing your surfaces. It will soften your geometry angles, while retaining a simple enough base mesh to work with, so that quads appear to have a smoother appearance.