There are 3 major portions to the graphics pipe-line. Modeling, Rendering, and Shading. You feed a model into a renderer, the renderer decides which pixels go where and what data to pass to the shader, the shader assigns a color to the pixel. There are three popular ways of modeling objects, polygons, patchs, and subdivision surfaces.
I would argue that these are really all the same method. Since Catmull-Clark subdivision surfaces are NURBS surfaces. And fundamentally, we model by pushing and pulling vertices around. They just have different effects on the surface. Furthermore, polygon models are just subdivision surfaces without smoothing rules.
Polygons are the most ubiqutious modeling representation. But they are also the most limiting. They define a totally irregular connectivity between vertices. Making the algorithms that affect them difficult and slow. This is in contrast to patches, which define a totally regular connectivity when they are possibly converted to polygons for rasterization. Unfortunately, they are hard to control, without significant algorthimic modification. I would argue that the sweet spot is a semi-regular connectivity which is provided by subdivision schemes.
Since, totally regular is a subset of semi-regular, we will ignore this conversion. What we are interested in is converting an irregular mesh to a semi-regular mesh. Fortunately, this problem has been researched and is called remeshing. What we need to do to remesh is remove unimportant vertices from the irregular mesh and replace them with regular vertices.
The removing of unimportant vertices is called simplification.