In the 1920s, American astronomer Edwin Hubble discovered that the universe is expanding. Not only are distant stars and galaxies moving away from us, but further stars and galaxies move away from us faster than closer ones. This is because of the homogeneous nature of the expansion. Every distance in the universe expands by the same multiplicative factor, and so an object that is twice as far from us moves twice as far in a given time.
Our modern understanding of cosmology is based on Einstein's general theory of relativity. Shortly after the big bang, the universe underwent an inflationary epoch in which the universe expanded exponentially. After the inflationary epoch is complete, basic principles of general relativity tell us that there are three broad types of possible "expansions" of the universe. Loosely speaking, an open universe is one which continues to expand forever, and whose expansion never comes to a halt. A flat universe is one which continues to expand forever, but whose expansion is constantly slowing down. A closed universe eventually stops expanding and then starts contracting until the universe ends in a "big crunch"—the opposite of the big bang. For many years, physicists and astronomers could not tell from observation in which of these three broad types of universe we live, but recent observations of supernovas have given us sufficiently precise information on motion that we now know that we live in an open universe. (The open universe simulated in this Demonstration is only one of many possible types, however, and is not a perfect simulation of ours.)
The numbers represent (randomly placed) clusters of galaxies in the universe; the blue "0" designates "our" location. As the universe's expansion is only visible on large scales the "0" encompasses the entire Milky Way galaxy which contains the Earth, plus some nearby galaxies.
The Demonstration starts centered on the Milky Way, and, after clicking "start", the other objects in the universe begin to recede. The original position of the objects are shown in gray, to make it easy to see how far each object has gone.
To see that the Earth is not a special point in the universe you can click on another object to re-center the Demonstration onto that object. This will pause the expansion so you can see the re-centering occurring without changing any distances between objects that are both black (current position) or both gray (original position), and then you will see that objects close to the new center have not moved as far away from the new center than further objects.
To scroll to objects which have scrolled off the screen, drag the mouse on an empty part of the canvas.
To "measure" the distance between objects, drag the ruler, or each endpoint of the ruler, to objects you wish to measure.
The "rescale the universe to fit" option changes the scale of the Demonstration while it runs so that objects never scroll out of the display. The effect is that everything in the universe gets smaller as the universe expands. Notice that this is true of the ruler, as well, so as the universe expands, the number of ruler-lengths between two objects increases. In fact, nothing in general relativity says that the universe expands—alternatively, everything in the universe could be getting smaller; there is no difference between the two descriptions.