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.
$1.$ Story of maths Part1 Part2 Part3 Part4
$2.$ Dangerous Knowledge Part1 Part2
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Should this be a community wiki? I really liked Beautiful Young Minds about the British IMO team.
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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
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Chaos: A Mathematical Adventure* (2010) and Dimensions** (2008) by Jos Leys, Étienne Ghys, Aurélien Alvarez, et al. are excellent, CC BY-NC-ND 3.0-licensed (i.e., free) movies about math.
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.
*downloadable from Archive.org, HD720p torrent, or HD1080 torrent
**on YouTube
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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.
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May not be fundamental maths videos, but I like the teaching style of Shai Simonson and his lectures in Discrete Mathematics and Theory of Computation are entertaining as well as enlightening.
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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].)
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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.
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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.
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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...
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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
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Flatland: Have you ever imagined if all of us live in a 1-dimension world? And after a long live in this one, the surprise to see the 3-dimension world in the first time?
Sphereland: the same as the flatland movie, but now, you live in a sphere
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These are some introductory videos from an 'old' (mid-80's) series co-produced by Caltech called Mechanical Universe. One on derivation and another one on integration. Features some nice Newton-Leibniz cosplay if nothing else. If you're into that sort of thing.
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Here is a (not well-known, it seems) video containing a lecture and interview with Richard Courant, former student of David Hilbert:
Göttingen and New York : reflections on a life in mathematics
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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.
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Socratica's youtube Abstract Algebra series is the best! Of the playlist, simple groups was my favorite.
I also like mathew salomone, mathdrbob and Michael Penn and everyone likes blackpenredpen.
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Seems like Richard Borcherds is making good use of the extra time we're all spending inside.
Just recently he started a youtube channel. So far he has given at home lectures on group theory, algebraic geometry, commutative algebra, and representation theory, as well as some interesting talks on assorted topics like monstrous moonshine, among other things.
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.