Unfortunately there is a pool of people that think the water is beneficial. If you were moving fish and media to a new tank i would say its worth chucking a 50/50 mix of old and new in if stocking straight away, if anything just to help settle the fish. For cycling, its pretty useless.
"Mature Media" can come in quite a forms, depending on the person donating it. It will pretty much be any type of media you can buy on the market, sponge, ceramic, floss, etc. This is where the majority of bacteria are. (Overtime a layer will build up on anything in the tank, a sort of slime, but this again doesnt provide the same level of biological filtration)
Some people might give you cuts from large sponges, some might give you a whole sponge from an external, etc.Some might give you a bag of ceramis balls or noodles, all are good. Especially ceramic. Not many people will give this away as its normally expensive, and lasts pretty much forever. Nothing wrong sponge however.
I normally give floss, as its the most commonly changed media i have. I swap out a quarter of an externals worth every 6-8 weeks, and by this time, its cycled, and the amount is probably good enough to immediately lightly stock the tank, and build up very gradually over a period of weeks. (Would always recommend a test with some ammonia over a few days before putting fish in however, which i can also place in self seal test tubes and post.)
In other words, only something from inside a filter is any good, and you get what you're given lol
Hope your parts turn up soon