The universe does not expand in space, instead it consists of expanding space. From this statement flows all the simplicity and complexity of modern cosmology. ~ Edward Harrison, Cosmology: The Science of the Universe
Ever since Edwin Hubble published the famous results of his painstaking observations, we have come to realize that we are living in a Universe that is expanding. Decades after his discovery, the misconceptions about what they mean remain, no thanks to some popular science literature (for which I will be forever indebted to for igniting my passion for cosmology!).
For one thing, the redshift (shifting of light waves to longer wavelengths) due to expansion is often confused with the Doppler redshift. Doppler redshifts have to do with relative velocities between the source of light and the observer. Expansion redshifts however, are caused by wavelengths of light being stretched as the light waves propagate through space that is expanding.
Also, galaxies are sometimes described to be rushing away from our own Galaxy. Although this was the initial interpretation of Edwin Hubble and other astronomers then, it has since been found to be untrue. In fact, it is space that is expanding. Some people imagine our universe expanding in space, or even have the concept that the Big Bang was an explosion in space and time – and therefore ask questions like: What happened before the Big Bang? Where in the universe did the Big Bang occur? What is the Universe expanding into? These absurdities disappear when we understand that space and time expanded out of the Big Bang – which was never really an explosion in the first place. There is no centre of the universe. The Big Bang happened at every point in space.