In the series "Mathematics for non-mathematicians", I recommend a talk by Dan Freed, titled "The Hodge Conjecture" - but you don't need to be interested in the Hodge Conjecture to benefit from this video!

You can find the slides here and a video (1:18h, Quicktime MOV format) here.

As I said, you don't need to be interested in the Hodge Conjecture (although it's one of the so-called millennium problems, and you would get a million $ for a proof or disproof). You'll get a glimpse on some central ideas of geometry from this talk, including:

• Projective plane (where every two lines intersect in a unique point, but maybe a point at infinity)
• Cycles (certain "shapes" in spaces, like points, lines, circles)
• Complex numbers (you don't need to remember i²=-1)
• Polynomial equations (like x²+y²=1)

At the end, it's a bit fuzzy about integrals, differential forms and the actual Hodge Conjecture, but if you weren't looking forward to learn anything about the Hodge Conjecture, this spares you about 30 minutes of the video.
If you're really interested in the Hodge Conjecture, take any book about Algebraic Geometry and learn some of what's in there. If you did this already - then what are you doing here?

When people ask me about the stuff I learn in mathematics, I try to explain something about homotopy/topology, circles defined by polynomial equations or projective space. This usually conveys the idea that mathematics is not all about number crunching or calculus. So this video is in the same spirit of explaining what mathematics is.

If you don't want to know anything about geometry (maybe because you hate mathematics since school), then I recommend this video (10 minutes, but you can skip the last 5 minutes):
"The most important video you'll ever see" [sic]