Subdivision surfaces represent an important alternative surface representation to simple polygon meshes. In fact, they provide an elegant means of connecting piecewise-linear meshes with piecewise-polynomial spline surfaces.
The following course notes from SIGGRAPH 2000 provide a very good introduction to some of the basic subdivision material.
Please look specifically at Chapter 2 (Basic Ideas). (Obviously, I don’t expect you to read all 200 pages!)
I recommend reading the rest of the Course Notes at your convenience. At the end, you’ll find a compendium of several subdivision papers.
Here are some other useful and interesting papers on subdivision surfaces:
T. DeRose, M. Kass, and T. Truong. Subdivision surfaces in character animation. In Proceedings of SIGGRAPH 98, pp. 85–94, 1998. [PDF]
P. Oswald and P. Schröder. Composite primal/dual sqrt(3)-subdivision schemes. CAGD 2003. [PDF]
D. Zorin, P. Schröder, and W. Sweldens. Interactive multiresolution mesh editing. In Proceedings of SIGGRAPH 97, pp. 259–268. [PDF]
D. Zorin and P. Schröder. A unified framework for primal/dual quadrilateral subdivision schemes. CAGD 2002. [PDF]