I'm also always struggling with these kinds of questions and I have essentially two answers for it.

If I carry a pen and paper and I feel like I have the time to explain stuff and want to go the less ridiculous (See my second answer) I talk about graphs and Eulerian paths. There is a childrens riddle known all across Germany called the "Haus vom Nikolaus"(House of Santa Claus) and it is the question whether some graph looking like a house has an eulerian path and almost everyone knows at least one Eulerian path. I then ask whether you can change the endpoint and the starting point. Afterwards I ask them whether the "Doppelhaus vom Nikolaus"(The double house of Santa Claus) i.e. two houses sharing a wall has an Eulerian path and if the audience is really interested I prove with them that this does not have an Eulerian path.

Nevertheless since I'm a topologists this is not really satisfying as it does not really represent my research interest. So at some point I came up with the following:

I take off my belt and close it again in front of the audience and declare this the "standard belt". I then twist the belt once to produce a Möbius band and ask whether we can play around with this to deform it to the standard belt. After playing around with it they are usually deeming this impossible but of course as a mathematician this is not the full answer so a proof has to be presented. Usually with a bit of help they count the boundary components or the sides and see that this is in fact not the standard belt. If I feel like the audience likes these kind of arguments I would then twist the belt twice and ask what happens now. So far I have never encountered a good answer to this, but after a while I explain to them how it is not possible in $\mathbb{R}^3$ and that the reason lies in the invariance of the winding of the normal bundle of the belt (explained in non-mathematical terms) and after all this I close with the final statement, that if we would live in $4$-dimensions we could untwist the two times twisted belt. If feel like this is quite an honest representation of the problems I'm actually interested in and people get an idea what I'm dealing with everyday.