I'm trying to find out more about geometry of surfaces and, in particular, Gaussian curvature. I understand that it can be defined in terms of the principal curvatures (extrinsically) and also intrinsically, and that the result that it can be equivalently defined in these two ways was a significant result. Does anyone know where I can find a nice, clear proof of the equivalence of these definitions and perhaps some historical/background information on their importance? Thanks in advance.
I like the presentation of the Theorema Egregium in A Comprehensive Introduction to Differential Geometry (Volume 2) by Michael Spivak. A translation of the original paper by Gauss and the historical background can be found in General investigations of curved surfaces of 1827 and 1825 by James Caddall Morehead and Adam Miller Hiltebeitel (the 2005 Dover edition).
It seems to me that the most difficult step is learning and understanding the intrinsic definition of Gauss curvature. The wikipedia page, http://en.wikipedia.org/wiki/Gaussian_curvature, provides a few different versions.
My favorite way to describe the extrinsic definition is the following: To calculate the Gauss curvature at a point $p$ on the surface: Move the surface so that $p$ is at the origin. Rotate the surface so that the $xy$-plane is tangent to the surface at the origin. The surface near the origin is now given as a graph $z = f(x,y)$, where the partials $ \partial_x f $ and $\partial_y f$ vanish at the origin. The second fundamental form of the surface at $p$ is given by the Hessian $\partial^2 f(0)$, and the Gauss curvature at $p$ is the determinant of the Hessian.
Proving the theorem egregium is now pretty straightforward, especially if you remember that you don't need exact formulas for anything but only second order approximations at the origin.
I guess you could have a look to the marvelous book A panoramic view of riemannian geometry by Marcel Berger. I do not have it there right now, but I am pretty sure it contains a discussion of the Theorema egregium, and it contains a lot of connections between the various subjects of Riemannian geometry.
Chapter 3 ("The Geometry of the Gauss Map") of Do Carmo's "Differential Geometry of Curves and Surfaces" has a very nice discussion along these lines, though I don't remember much historical background being present. For that, you might try Struik's "Lectures on Classical Differential Geomtry", which includes historical remarks on most of the main results.