Who hasn't heard of the Mobius strip? The Mobius strip is the thing you get if you take a strip of paper, twist one of its ends half a twist, and tape the two ends together. It has the incredibly counterintuitive property that it only has one "side" -- if you were an ant, say, walking along the surface of such a strip, you could walk and walk and walk and suddenly you'd be where you were before but on what you'd think of as the "other side".
The article linked to -- which seems to be derived from a press release as I've found it in a few other places -- says that:
Since 1930, the Moebius strip has been a classic poser for experts in mechanics. The teaser is to resolve the strip algebraically--to explain its unusual shape in the form of an equation.
This seems a bit surprising -- it's easy to write down an equation that parametrizes the Mobius surface as basically a thickened circle. If you go to nature.com's writeup of it, though, it begins to make sense. What it actually means is that if you build a model of the Mobius strip out of an actual piece of paper or some other foldable material, the actual shape it will take is difficult to predict. This is what Eugene Starostin and Gert van der Heijden have actually done.
The difference here, then, is that when I hear "Mobius strip" I think of some sort of Platonic object, floating there in space, which could be realized in any material and is not subject to the laws of physics but only the laws of geometry, whereas the vast majority of people picture a piece of paper which has been twisted.
By the way, you can read some quasi-mystical stuff about the Mobius strip at Mobius Products and Services. As far as I know the Mobius strip has nothing to do with the Mobius transformation other than that the same person came up with both of them. But the legitimate journalists are subject to this mysticalism too -- nature.com points out that the Moebius strip, when viewed from a certain angle, looks like ∞, the sign for infinity.