(a) What is a Mobius strip? You could read about it on MathWorld, but it is preferable to make a Mobius strip out a strip of paper and see for yourself why it is so bizarre. You need to tape the ends together after making a half twist in the paper. Then colour one side of the strip red (or another favourite colour) and observe that, in fact, the Mobius strip only has one side. Cool, eh? It also only has one boundary edge; convince yourself by running your finger along one edge and arriving back where you started after travelling around the whole boundary.
(b) What is a Klein bottle? To make the Klein bottle, start with an open cylinder and then paste together the ends with a half twist, identifying opposite points on the boundaries. This is easiest to do in four dimensions. In three dimensions it has to be understood with a pesky self-intersection. You can see that the inside is really the outside; the Klein bottle has no interior. It actually has zero volume. Also, it has only one side, since if you trace with your finger all over the surface you will find that you can reach all of the surface.
(c) What is the joke? I really want a Mobius scarf and Klein bottle hat, and anyone can buy it for me. Choose any one at all, since, you know, one side fits all!
(d) Is that really the funniest maths joke of all time? Are you kidding?! I'm rolling on the floor laughing. There are other funny ones, of course. Would you like to hear some more another time? :)