Though I am not teaching on a regular basis, I often explain what is mathematics to laymen. My explanations tend to converge toward the following lines:
1) Mathematics is poetry: 2 quotes:
Quote 1 : Mathematics is the art of giving two names to the same thing AND the same name to two different things ( HENRI POINCARE).
Quote 2 : In mathematics you have an absolute liberty, the price to pay for this is that you have to be very precise ( YURI MANIN).
2) Maths is made of observations and rendering them with an eventual need to make up a new language, just as anybody would need one in a complex and professional field (say dancing).
3) THE WORLD OF A SURFACE IN ITSELF
Then show them a band a paper,make it a cylinder (2 faces, 2 circle boundaries) . Then link it with a twist and ask them to count the boundaries and then the face. ( They will be astonished and see the difference for themselves ...)
3.a) On a Moebius band a river drawn in the middle ( a blue pencil will do) has only one bank( Let them check it), it is a different world if you live on it ( you are little bugs with no sense of the third dimension)
3.b) Tell them about the game of cylindrical chess (played on a torus, abstracting the game if necessary ( just moving pieces) those who know the rules of chess feel better. Show them the game while remaining flat, then tell them that for someone really dumb you could imagine to produce a real torus by bending the board and using magnetic pieces. As a world (a fighting one say) you may observe that proximity is changed.... (chess serves as a surrogate topology in this but you do not have to pronounce the frightening word topology).
3.b) Back on the Moebius band : the game of chess on it is not the same as the cylindrical one: a piece does not move the same way and does not aspects the sames cells...
4) DIFFERENT WORLD AND VIEWS :
Now take a band make it a cylinder with a knot first ( need a band long and thin enough) put it side by side with the normal cylinder and ask them is it the same.
After their answer your is of course yes AND no (the ambient or the embedded surface) .
A matter of point of view.
5) USEFULNESS OF MATHEMATICS:
Of course there lots of applications but the killing example is : In 400 BC Greeks were doing land regrouping (consolidation),each land was measured by willing geometers. A year later there was plenty of lawyers at work because the pieces of land had been measured by perimeter!!
Tell them that it might seem obviously stupid to do so, yet basic school told them about surface concept. Moreover using the perimeter might be a good way to do things if the goal was not farming but showing off with high flags and poles. Again many points of views blablabla...
NOTE : The interactivity is essential at least when checking the boundary of MB with the finger or sight for some. This is a close call for ten minutes, part 5 can be removed.
Try it on some none mathematical friends first after three times you will be probably quite sleek. It is also important to have the right length and width for the band paper 12 inches by less than one roughly usually the side of a sheet of paper...