For those of you who weren't in Siberia today, here is a video of the entire totality, when the sun was completely covered by the moon. The video was taken by Ivan Komarov of Manjerok, Siberia, and sent to cnn.com.
The video starts in the final seconds before totality, when the last bits of the sun are disappearing behind the moon. This is often called the "diamond ring" effect, as one brilliant sliver of the sun shines through a lunar valley.
In the instant the sun is completely covered, the sun's outer atmosphere, or corona, comes into view. What you are seeing is a very tenuous, very hot (millions of degrees!) gas. The light we see is not from the gas itself, but light from the sun's surface (called the photosphere) being reflected toward us. The sun's corona is always there, but is completely obscured unless something (like a 2000-mile diameter rock) blocks the sun's photosphere.
After about 2 minutes, you can see the diamond ring happen in reverse. Notice how fast the sun gets bright again! Its photosphere is so brilliant, that even the tiniest sliver overwhelms cameras (and can damage your eyes). During the totality, it is completely safe to look at the sun without protective equipment, even through telescopes and binoculars. But the instant the photosphere re-emerges, you must use solar filters again, or you will permanently damage your eyes.
Unlike lunar eclipses, total solar eclipses have scientific value, though not as much as they used to. Solar astronomers flock to total eclipses with all sorts of expensive equipment for two minutes of intense data collection. With the sun's photosphere out of view, astronomers can study the very outermost layers of the sun's atmosphere from the ground. Before satellite telescopes, total eclipses were the only time this could be done! But, these days, satellites can use fancy optics to create their own eclipses whenever they want, so the science value of a given eclipse has gone down somewhat. But, it is far, far cheaper to run new experiments on the ground during a total eclipse than to launch a satellite, so there is still valuable and groundbreaking science that is done during total solar eclipses.
Total solar eclipses were also the first independent test of Einstein's General Theory of Relativity. Einstein predicted that the sun's gravity should bend the light from stars ever so slightly; careful measurements of stars (which become visible during totality!) taken during eclipses found that Einstein's predictions were spot on. Here is an article discussing some of those first historic measurements. It is important to remember that this was a pure prediction -- nobody had the slightest idea that gravity could bend light, so no other theory predicted any bending, and Einstein's theory got not just the bending right, but the the amount of bending right. It was an amazing prediction, and an even more amazing confirmation of Einstein's theory.
As I said, satellites can now make artificial total eclipses whenever they want; this has led to a revolution in understanding how are sun works, and how it interacts with the Earth. The SOHO spacecraft releases images and movies of the sun and its corona daily; go here and click on the image to look at, or the link to movies (at the bottom of the page). The "false eclipses" are the pictures from the "LASCO C2" and "LASCO C3" instruments, where the white circle indicates the size of the sun. The central dark area (much bigger than the sun) is a disk used to block the sun's light. Right now, you can see the planet Mercury passing the sun. And this page has movies from throughout the SOHO mission's storied life.