What is a "Region" in Mathematica 10?

I think this question is broad since Regions are used in various fields for various purposes. Think computational geometry (Voronoi diagrams, Delaunay triangulations, Convex hull etc.), finite element methods (PDEs), solid geometry etc. In addition to the links provided by Mr.Wizard, here are some additional links that might get you started:

Solving Partial Differential Equations with Finite Elements

Element Mesh Generation

Element Mesh Visualization

Finite Element Programming

I am not sure what you are looking for but in case any of these are new to you:

• New in Mathematica 10: Geometric Computation

• Guide: Geometric Computation

  • Mesh-Based Geometric Regions

  • Region Properties and Measures