A Gömböc (pictured right) is a mono-monostatic body: it has exactly one stable and one unstable point of equilibrium. In other words, if you set it down on a flat surface in any orientation, it will automatically reorient itself to its upright position.
So it's just like a Weeble, then? Well, not quite: unlike a Weeble, a Gömböc, by definition is convex. But more importantly, and very unlike a Weeble (which embeds an off-center heavy mass that acts as the self-righting mechanism) a Gömböc is of uniform density. You could carve a Gömböc out of glass or marble or metal and it would still maintain its self-righting property. But it would have to be very pure marble, and you'd need some fine engineering skills: if the shape deviates by more than 0.1mm for 10cm of size, it doesn't work.
The video below, from the UK show QI (of which I've only ever seen clips -- anyone know if you can get whole episodes in the US?) shows a Gömböc in action, and also its inventor/discoverer, Gábor Domokos.
The Gömböc doesn't have any real practical application (other than the suggestion that this kind of shape was evolutionarily selected for turtle shells, so they could self-right) but it's just awesome and I want one. But it's a lot of money to spend on a cool paperweight, and if you drop it, you can't fix it.