Media is what is put in the filter to maintain water quality. In the Aquaclear 500/110, when you buy the filter, it comes with biomedia (Biomax), mechanical media (the sponge), and chemical media (the carbon).
Very briefly, cycling is the process of establishing bacterial colonies in the filter that will break down and convert ammonia (caused by nitrogenous waste products and which is harmful at high levels) into other less harmful/relatively harmless nitrogen compounds---first to nitrite and then to nitrate.
This may help you understand cycling and cloudy water better:
http://www.honors.montana.edu/~weif/fir ... ater.phtml