Blender - 3D Meshes - Why must they be made with (mostly) "Quads" ?

Quads allow you to better subdivide your model for higher detail and provide for cleaner animations/deformations. Using triangles during the modeling phase will sometimes cause surface distortions in the model that quads do not produce. Also, adding edge loops cannot be done with anything other than quads if you want to manually subdivide your model for more detail.

There are a few reasons (and anybody feel free to correct me if I'm wrong):

1. Quads are easy for your GPU to then break down into triangles, which makes it much easier for the GPU to compute things on it.  You have 3 points, and that makes a plane, which then you can do things like calculate shaders.  Anything above quads (called generally n-gons, more than 4 sides) is hard for your GPU to break down into triangles and do the computing it needs to do.  
2. Because of compounding problems from n-gons and even n-poles (verts with more than 4 edges joined), you'll have a harder time later in the pipeline for things like skinning and animation.  Things will deform in wonky ways, ways that'll take up time trying to fix than to do right in the first place.  It kind of compounds exponentially, e.g. catch something early, it's easier to fix.  Catch something late, it's going to take a lot of time to fix.  Really, it's just better to do it right as early as possible.
3. I think quads are a little easier to display than triangles for humans to understand what is happening to the mesh, as well.  Sure, we could do it in triangles, but it's easier to see how the light is falling with quads, I think.

I'll add to this that sub-d modifier works better with quads, with tris/poles/n-gons you might get wonky results, also this come back to animations and deformations.

However, once you know all this and have experience, you can break this "rule" like any other and use it to your advance. Those "wonky" results might be exactly what you need for the project you are workin on. 

remember, what matters is the end result/use. 

Thanks everyone, after your explanation(s) it now makes perfect sense to me.

silentheart00:
Thank you for reminding me to the background on how GPU's like digest a 3d scene.

hee hee hee. Because Jonathan said so....

In the beginning, Jonathan created quads. And he saw that it was good.

In all seriousness, these guys pretty well hit it on the head. It sounds like the animation of said non-quads becomes problematic at best. That's the biggest thing I took from everything I've heard or seen on the site.

No problem!

I've just finished Jonathan Lampel  new Mesh Modeling Bootcamp course and I now have no quads questions anymore. Thank Jonatham L. for educating the Quad-Mindset into my CAD ( and vertix_only) -MindSet.

something I might add to this fine list of reasons is a simple idea of the loop tools in Blender...

in order for the loop tools (edge loops or face loops) to work correctly quads are needed...

if geometry is a triangle or an ngon there is no direction of North, South, East, West...

without this direction the loop tools come to an end (terminate) because there is no continuing without knowing which way to go...

this is important especially to box modeling when you want to be able to add loop cuts and not end up with spiralling loops or loops terminating in some odd place...

Thanks Mark for your explanation.