A Möbius strip is a paper strip with one twist, glued end to end — a "one-sided surface" with no separate front and back. A Klein bottle takes this idea one step further, so that the distinction between inside and outside disappears entirely. Draw a line on it and fill it with water to see for yourself.
A Möbius strip is made simply by twisting a long paper strip once and gluing the ends together. But strangely enough, it becomes a single connected surface with no separate front and back. A shape like this is called a one-sided surface.
This property is put to real use, too. Building a conveyor belt in the shape of a Möbius strip lets both sides of the belt wear down evenly, making it last longer, and the familiar recycling symbol also took its idea from the Möbius strip. The Klein bottle in the second tab below takes this idea one step further.
Turn on "draw mode" and trace a continuous line along the surface. Keep drawing without lifting your finger, and you'll see the line eventually connect to what you thought was the opposite side.