Here's a rather concise explanation I think...
Albert Einstein's General Theory of Relativity, which established the relationship between matter, space, time and gravity, governs modern cosmology's view of the universe. But when Einstein began to apply his theory to the structure of the universe, he was dismayed to find that it predicted either an expanding or contracting universe--something entirely incompatible with the prevailing notion of a static universe. In what he would later call "the greatest blunder of my life," Einstein added a term called the cosmological constant to his equations that would make his calculations consistent with a static universe.
Einstein admitted his mistake in 1929 when Edwin Hubble showed that distant galaxies were, indeed, receding from the earth, and the further away they were,the faster they were moving. That discovery changed cosmology.
Enter Hubble's Law.
The familiar sound of a train whistle as it recedes into the distance is a consequence of the Doppler Effect. As the train moves away from the listener, the crests of the sound waves are stretched out or shifted, resulting in a lower pitch. The faster the train recedes, the more stretched out the waves become. The same holds true for any wave-emitting object--whether they be sound waves, light waves, or radio waves. Conversely, the wavelength of objects that are moving toward us are shorter than those emitted by an object at rest.
Atoms emit or absorb light in characteristic wavelengths: hydrogen, helium, and all the other atomic elements have their own spectrum signatures. In the early part of this century, Vesto Slipher was studying the spectra of light emitted from nearby galaxies. He noticed that the light coming from many galaxies was shifted toward the red, or longer wavelength, end of the spectrum. The simplest interpretation of this "redshift" was that the galaxies were moving away from us.
Hubbles's Law
Hubble, who had been the first to establish that the universe included many other galaxies outside of our own, noticed something else: the galaxies were receding from us at a velocity proportional to their distance. The more distant the galaxy, the greater its redshift, and therefore the higher the velocity, a relation known as Hubble's Law.
http://http://archive.ncsa.uiuc.edu/Cyberia/Cosmos/Images/hubble_lg.jpg
The velocity v could be determined by multiplying the distance R by H, the Hubble constant, given by the slope of the line in the above graph, in units of kilometers per second per million light years. The Hubble constant describes the universe's rate of expansion.
The apparent linearity of Hubble's Law implies that the universe is uniformly expanding. What does that actually mean?
For one thing, it means that no matter which galaxy we happen to be in, virtually all of the other galaxies are moving away from us (the exceptions are at the local level: gravitational attraction pulls neighboring galaxies, such as Andromeda and the Milky Way, closer together). In other words, it's not as though we here on earth are at the center of the universe and everything else is receding from us. The universe has no "edge" as such.
