Via Uncertain Principles i found this one minute explanation of why we think there is dark matter. This is a video of a Scientific American editor telling us that stars in galaxies do not rotate as crumbs in a cup of coffee, but as the text on a cd disk.
It's always easy to be a besserwisser when it comes to your own field of research, but I'm going to give in to the impulse to try my own explanation since I've made a promise to myself to write more about science. It is not strictly correct that the stars in a galaxy move as if they were attached to a solid disk, and I would be happier with a demonstration that did not introduce that notion.
Here follows Åka's (really short) explanation of why we think galaxies contain dark matter:
When we measure the velocities of stars in galaxies, we find that the stars far from the center move too fast. The gravitational force caused by the matter we can see is not enough to hold stars with those velocities bound to the galaxy. There is a well known relationship between the velocity in an orbit and the pull of the gravitational force, known since Newton. Therefore we conclude that there must be more mass, and more gravitational force, to hold the stars to the galaxy. This excess mass that the visible stars can not account for is the mysterious dark matter.
I still have to think of a way to demonstrate this with things found in my office ;)
Why we know that this invisible matter is not just dust or clumps of ordinary atoms (the "baryonic matter" mentioned in the video is just physics lingo for atoms, or more stricly atomic nuclei) is a different question. Maybe I'll return to that another day.