Mobious strips are miracles of topology. It's a surface with only one side and only one boundary.
Essentially, when you give a loop a simple twist and reconnect it, it joins the right side to the left side. When you cut it straight down the middle, then unlike everything else in this world, you don't get two objects — you get one continuous strip with a two full twists in it.
If you start with two twists, for a full rotation, and then cut it down the middle, then you get two loops hooked together.
A great magic trick you can do is to tell people that the hand is quicker than the eye and that they should watch carefully to see how you reconnected the ends of the strip back together after you "separated" them. The magic, of course, is that there is no magic — simply something counterintuitive.
The strip ends can be connected with tape or staples. You can connect them together in series to get a very long strip, which works better if you've made your strip really wide.