The media you refer to is what fills up the insides of you filter and comes in lots of different forms. The main ones are sponges, ceramic tubes, bio balls and floss. All of them do slightly diffrerent jobs within your filter but what they usually have in common is a large surface area over which good bacteria colonise. These bacteria which look like browny green slime and gunk then deal with your fishes' waste as water passes over them.
So matured media is basically sponges, ceramic tubes etc. which come from a well established tank. They are useful in kick starting a brand new filter because they already contain the good bacteria needed.
There are many other things which can be put into your filter to fulfill different roles and this article provide more detail: