I know of two good mathematics videos available online, namely:
Do you know of any other good math videos? Share.
I have compiled a list (1500+) of math videos at http://pinterest.com/mathematicsprof/ . If anyone is aware of others, please send them to me.
77 instructional videos on category theory:
I know you said "only one video per post", but I'm not posting 77 times...
Most of the talks at MSRI are videotaped and placed on the web here:
My personal all-time favorite is the Klein Four with their song "Finite Simple Group (of Order Two)"... it has lots of puns on topology in it, but I guess it doesn't teach anything.
I believe this was mentioned elsewhere, but for completeness, here's Serre on writing.
This video about Andrew Wiles and the proof of Fermat's Last Theorem is the only time I've seen the real excitement of mathematics presented accurately.
'Not Knot' is also a nice vid
At the accessible end of the scale, Vi Hart's "doodling in math class" series and subsequent videos are a delight.
MIT's OpenCourseWare has a few math courses up:
GRASP is a new lecture series at the University of Texas at Austin, which is aimed at bringing some of the fundamental concepts and big picture of the GRASP areas (Geometry, Representation, and Some Physics) to a wider audience (the intended target audience are beginning graduate students).
The Institute for Advanced Study tapes some of its lectures. They tend to be very good.
You probably won't learn much actual math from it, but One Geometry is funnier and catchier than a Snoop Dogg parody about 3-manifolds has any right to be.
Along the sphere eversion lines, there is also the energy-minimizing sphere eversion constructed by Rob Kusner. I think there is a video of it at http://www.youtube.com/watch?v=I6cgca4Mmcc, though it isn't labelled as such.
Rob also has written a paper about the history of the minimax eversion.
My personal favorite in Dimensions, that was mentioned before by Gerald Edgar. For a neat and clear exposition the Geom.of 3 manifolds, Poincaré conjecture, etc I recommend this lecture by C.McMullen. Or Das Schöne denken (hosted at the HIM in Bonn), for a good "glimpse in the world of the mathematician". Jos Leys' mathematical imagery contains some (interesting) videos and (a lot of beautiful) images.
There are Stephen Boyd's lecture videos on convex optimization:
I guess all of John Conway's lectures are great. Some of those can be found here : http://www.math.princeton.edu/facultypapers/Conway/
Not lecture videos or anything, but the stuff from Oliver Labs is very good for just illustrating geometric stuff, like blowups and dual curves.
The series of videos from IAS School of Mathematics
Dror Bar-Natan has begin putting many of his lectures and talks online in video format. I'm not claiming that these are the 'best' online maths videos, but they're certainly interesting, and in particular he's come up with some neat tricks to associate publicly editable annotations with particular moments in the video.
I am surprised that nobody mentioned the four-week workshop at Göttingen on arithmetic geometry in 2006 summer. Almost all of the videos are still available. Wonderful videos.
This is an old thread, but this video was recently posted to the Don Davis topology list, and I have to share it. It was created by Niles Johnson at UGA and it illustrates the Hopf fibration.
This one is quite old but it was fun when I watched a few years ago. It's about Fermat's Last theorem.
Lots of Lie Theory talks: http://sms.cam.ac.uk/collection/533438?mediaOffset=20&mediaMax=20&mediaOrder=asc&mediaSort=title#Media
Videos recorded at IMPA:
(some in English, some in Portuguese)
John Stillwell - ET Math: How different could it be? A nice talk given at the SETI Institute.
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.