You supply the red shift value (z) that you're interested in, and certain additional parameters (for which the best estimated values are supplied as defaults), and the calculator will tell you when the light was emitted (in billions of years since the big bang), how long the light has been traveling, and several other interesting but more arcane quantities. Links to brief explanations of many of the variables are provided.
By varying some of the parameters (such as Hubble's constant, H0, and ΩM) you can play with different cosmological models.
If you want to find a red shift value that corresponds to some given light travel time, you have to make a guess (e. g. z=3), plug it in, and then vary z until you come close to what you wanted.
If you can't quite recall what all the terminology means, Wright even has a very good tutorial on the whole subject of cosmology. Or perhaps instead, or in addition, you might look at my cosmology turorial.
And if you're really serious about cosmology and would like to know in excruciating detail what the mathematics are behind the calculations, take a look at Wright's paper at the arXiv: [astro-ph/0609593] A Cosmology Calculator for the World Wide Web
Oh, and if you're not quite sure why anyone in their right mind might care about this sort of thing, there's a nice Scientific American article by Abraham Loeb that was just published: The Dark Ages of the Universe. It deals with the period of time from when matter and energy decoupled – the time of the cosmic microwave background – up until the first very bright stars began to shine. The times correspond to a redshift around z=10. The article explains what's so interesting about this period.
Tags: cosmology, red shift, cosmology calculator