This is an offshoot of the question on Fun math outreach/social activities. I have listed a few videos/documentaries I have seen. I would appreciate if people could add on to this list.
Chaos introduces determinism vs. indeterminism, the butterfly effect, and Lorenz attractors. Dimensions does a very good job introducing stereographic projection, fourth dimensional geometry, complex numbers, fibrations, and proofs.
Both films are dubbed and subtitled in various languages.
Documentary of Paul Erdos "N is a Number" is another good documentary. Also there is a documentary of Srinivasa Ramanujan called God, Zero and Infinity was released by TIFR in 2009.
Another documentary named Hard Problems was released by MAA which shows how US students performed in the IMO.
When I was a kid we used to watch "Donald in Mathmagic land" starring Donald Duck. While in undergrad we all gathered to watch and it was still enjoyable.
Here's a really excellent video, Finite Simple Group (of Order Two), that's definitely more towards the fun side than educational, although if you're not getting some of the jokes it may inspire you to look up a thing or two.
Should this be a community wiki? I really liked Beautiful Young Minds about the British IMO team.
Bob Gardner's page is a treasure trove.:)
Here is a (not well-known, it seems) video containing a lecture and interview with Richard Courant, former student of David Hilbert:
A Russian documentary about Grigori Perelman and the Poincare conjecture. Now with English subtitles!
There was some great answer on this post. But I preferred to share my answer.
As mentioned before Dimensions videos are a very good example but it's somehow boring because of it's slow rate of speaking.
Also there is some movies about math which can be helpful such as a beautiful mind, the man who knew infinity , Fermat's room, ...
But what I highly recommend :
$2)$ some youtube channel about math like
- Numberphile [my favorite sections are choosing toilets, elliptical pool, shapes and hooks, How they found the World's Biggest Prime Number, ...]
- Vihart [my favorites are Doodling in Math Class Connecting Dots, Hexaflexagons, Möbius Music Box, Wind and Mr.Ug,...]
- Mathologer [my favorites are Always WIN with math, Rubik's cubes, gyroscopes , ...]
- $3$Blue$1$Brown [my favorites are Euler's formula with introductory group theory , Visualizing the Riemann zeta function ,...]
- Henry Segerman's videos [my favorites are Stereographic projection, Illuminating hyperbolic geometry ,... ]
Nova's Mathematical Mystery Tour is still on youtube (let's hope it stays there): (link to first part): http://www.youtube.com/watch?v=vPSx2CkKE3c
I enjoyed Between the Folds, which gives a rather high-level view of the mathematics behind origami and how they provide means for constructing incredibly complex shapes.
A few years back the NHK made a documentary called "The Spell of Poincare Conjecture". (You can find a possibly non permanent link here to part 1 of 4 [with also links to the rest of the video].)
For youngsters, there is Donald in Mathmagic Land.
Fractals - Hunting the Hidden Dimension by PBS: Nova.
On Srinivasa ramanujan the genius mathematician from India. I believe a hollywood movie based on his life is being made, with Madhavan playing the role of Ramanujan.
Ramanujan:Letters from an Indian Clerk http://www.youtube.com/watch?v=OARGZ1xXCxs
A very nice TED talk by Robert Lang about the mathematics of origami: http://www.ted.com/talks/robert_lang_folds_way_new_origami.html. It goes very well with this Wikipedia article: http://en.wikipedia.org/wiki/Mathematics_of_paper_folding. The talk itself isn't heavy mathematics, but the mathematics behind the talk is very interesting.
I found this talk about the life and work of Évariste Galois Superb: here is a link
you all may want to see BBC: Code Breakers Bletchley Parks lost Heroes http://www.youtube.com/watch?v=JF48sl15OCg
Documentary about the story behind the German cryptography systems used in World War II that gave birth to the digital age i am not talking about Enigma but an even tougher system, which Hitler called his 'secrets writer'.
Its story of and Bill Tutt and Tommy flowers whom i believe to be the inventors of worlds first computer.as the transcript goes "This Is the story of a secret war , and how two men changed the world and then disappeared from history"
If you are interested in cryptography this documentary clearly explains how ciphers works, XOR ciphers in particular .
While it's been a while since I've seen it, the old Geometry Center videos - especially Not Knot - are personal favorites. The Geometry Center's video page is at http://www.geom.uiuc.edu/video/ - as for Not Knot itself, I'm sure it should be findable easily enough...
I just found this amazing website Mathtube.org. Here is excerpt from the site's About page:
"What is mathtube.org? mathtube.org is a new multimedia resource hosted by the Pacific Institute for the Mathematical Sciences. mathtube.org gives users easy access to mathematical seminar and lecture materials including videos notes and slides. mathube.org allows users to view slides and video simultaneously, making it convenient to follow a lecture at your own pace and review anything you have missed."
Mathtube.org has dozens of lectures, for example there is a lecture by Ben Green titled On the Sylvester-Gallai Theorem.
I am surprised no-one seems to have mentioned the Numberphile youtube channel.
Meant for kids, I watched it as a kid, but I still love it as an adult:
Mathnet, here's an example: https://www.youtube.com/watch?v=FngBiPt7yRU
It's like a math version of the police show Dragnet. Lots basic math, deadpan humor, and bad puns. It's a few decades old though, so don't expect any great filming. Just a great show though.