I know of two good mathematics videos available online, namely:
Do you know of any other good math videos? Share.
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The "Touching Soap Films" series by Springer. about minimal surfaces. Some excerpts of the video are available here: http://page.mi.fu-berlin.de/polthier/video/Touching/Scenes.html
This video is less about mathematics, but about a fascinating mathematician in two bodies who helped saving medieval unicorns - students liked it.
My good friend Professor Elvis Zap has the "Calculus Rap," the "Quantum Gravity Topological Quantum Field Theory Blues," a vid on constructing "Boy's Surface," "Drawing the hypercube (yes he knows there is a line missing in part 1)," A few things on quandles, and a bunch of precalculus and calculus videos. In order to embarrass all involved, he posted the series "Dehn's Dilemma" that was recorded in Italy last summer.
On this page of sample animations using the k3d program there's a short animation of a "flower" blooming which is actually the first part of the sphere eversion.
For a course on cluster algebras (by S. Fomin): http://qgm.au.dk/video/mc/cluster/
EDIT: Some graduate short-courses in FCEyN, UBA, Buenos Aires, Argentina:
Here are the links to the videos of these 4 lectures.
Among the best math videos can be found here: http://www.khanacademy.org/
(or the youtube-channel: http://www.youtube.com/khanacademy )
There is everything from counting to solving differential equations with Laplace transforms - nearly 1.000 videos altogether (and the guy is funny :-)
Ken Ribet's introductory lecture on Serre's modularity conjecture. Useful and quite easy to follow and understand. http://fora.tv/2007/10/25/Kenneth_Ribet_Serre_s_Modularity_Conjecture
NMU(http://en.wikipedia.org/wiki/Independent_University_of_Moscow) and MIAN lectures 2009-2010 (in Russian)
Timothy Gowers' "The Important of Mathematics" never fails to instill a sense of purpose in my work, even when I feel I'm doing "useless" mathematics.
Richard Feynman gave the 1964 Messenger Lectures at Cornell University --- this is an endowed lecture series to which a number of famous scholars have been invited, including several physicists. His lectures were recorded, and Bill Gates bought the rights to them and has provided them to the public for free.
The content is mostly designed for a general audience, so if you have never learned physics you will learn something. And if you have studied plenty of physics already, you will be pleased to see the master at work in his prime. I very much enjoyed watching it.
Sir Michael Atiyah: Beauty in Mathematics.
The Abdus Salam International Centre for Theoretical Physics has lots of lectures in mathematics and physics.Some of them are difficult to find in other places(Complex Analysis,Abstract Algebra,Topology,Functional Analysis,Algebraic Geometry..).For the same topic(ex:Complex Analysis)there are lectures by 2 ore more lecturers so you can choose. http://www.ictp.it/ http://www.ictp.tv/diploma/index07-08.php?activityid=MTH http://www.ictp.tv/diploma/index08-09.php?activityid=MTH
A few talks under the heading "What is ..." (",,," could be "Morse Theory", for example) given at the Freie Universität Berlin can be found here:
Excellent lectures by Norman Wildberger on topics including: Geometry, Algebraic Topology, Linear Algebra, Foundations of Mathematics, and history of Mathematics
I am quite surprised to see Dan Freed's lecture of Hodge Conjecture has not been mentioned. (Although it is an old thread I believe this should be in here. Before there was a QuickTime video but I am grateful to find that it has been youtubed.)
As of today, the digitized tapes of CBMS Lectures on Probability Theory and Combinatorial by Michael Steele are online. I heartily recommend them — the style is informal, but educating: there are jokes, juggling lessons, speculations about the stock market, and all of these amidst beautiful mathematics.
Two recent videotaped lectures by Doron Zeilberger.
The complete introductory course on Algebraic Geometry by Miles Reid is very interesting (28 lectures following and extending his own undergraduate book on the subject), and his other set of lectures on Algebraic Surfaces.
A nice introduction to representation theory of compact lie groups, sl2(R) and other topics: http://www.math.utah.edu/vigre/minicourses/sl2/schedule.html
I found the Graduate weekend repository of lectures at the Mathematics Department of Duke's University very entertaining. There is more in the other folders(G.Tian, Langlands, just to name a few )$\ldots$
Some talks on history by some leading mathematicians (mostly in French):
The University of New South Wales in Sydney has an eLearning channel on YouTube that contains lectures on a number of topics, including Algebraic Topology, Calculus, and Linear Algebra. Some computing and engineering topics are covered as well.
Searching for a video relating to another question, I found this: My Calculus Project
David Cox's lectures in toric varieties at MSRI
Something really good to end the evening with :)