Vi Hart has many, many awesome math videos. Here's just one to get you started. OK, I can't resist linking to this one too. Very hard to pick a favorite.
Donald Duck in Mathmagicland. This is from 1959, so parts of it are a bit dated but it had a significant influence on me when I saw it sixth grade. I know many other friends and family who remember it fondly as well - a real classic.
A video from 1988 from Caltech, very well done: http://archive.org/details/theorem_of_pythagoras.
Möbius Transformations. Short but very well done.
Niles Johnson's Hopf Fibration video. Hopefully Niles will continue making more videos like this.
Not Knot part I and part II. This is a window onto some really advanced mathematics, but I think its worth showing to just about everyone. Perhaps more than any other video I know, it visually conveys one of the many amazing structures of modern mathematics. Its a good antidote to anyone who thinks that math=arithmetic.
Dimensions is a 9-part series of short movies that have excellent animations. Of course some episodes are better than others.
There are some other series which are more educationally oriented (and generally less visually stunning) but worth checking out: the Mathalicious project, the TED-Ed math section, and the Numberphile youtube series.
Finally, two movies which are good are "Between the Folds" (on origami and mathematics), and the BBC Horizons documentary "The Proof" which is about Wiles' solution to Fermat's Last Theorem. These are both about an hour, so they require a longer attention span than the ones listed above.
If you have any favorites which aren't listed here, let me know.
