Watch Outside In. (something we should all do anyway, to commemorate Bill Thurston's passing).
To understand the mathematics behind sphere eversions, you should first get a good intuition for the concepts of immersion and regular homotopy. I recommend Guillemin and Pollack's "Differential Topology" book for starters.
To see why it is not possible to turn a circle inside out, you should read up on the Whitney-Graustein Theorem. It basically boils down to the calculation $\pi_1(S^1)=\mathbb{Z}$, once you noticed notice that the normalized differential of an immersion $S^1\looparrowright \mathbb{R}^2$ is a map $S^1\to S^1$.
You'll find a few more resources related to sphere eversions on my web page.
