This is how I [a regular Joe] understand it:
There's objects in the universe that serve as http://en.wikipedia.org/wiki/Standard_candle" . Two are the most common used to calibrate astronomical distances: Type IA Supernovae and Cepheid variables. If you know how bright something is, you can estimate it's distance using an inverse square law of light propagation. Then.. you have to calibrate your measurement using something called "redshift".
Light propagates as a wave (as well as discrete packages of energy: photons). Being a wave, it has a measurable frequency and this frequency can vary for a variety of reasons. In this particular case, Astronomers measure what's called "redshift". Redshift is equivalent to a frequency elongation which can be estimated using the Doppler effect for light.
Analogous to the Doppler effect for sound, light waves can elongate (redshift) or compress (blueshift). The elongation in light from deep deep outer space is caused by 2 things: the speed of source and the expansion of the universe itself.
If you're not familiar with the Doppler effect, think of it as what you hear when you're standing still on the side of the highway and big loud truck passes by [the sound pitch will increase as the osingbject gets closer].
EDIT: Added first paragraph. It didn't pass the first time I posted