Ammonia is converted to nitrite by one type of bacteria, then another type of bacteria consume the nitrite and produce nitrate. In a cycled tank with some source of ammonia, nitrates will continuously increase as ammonia is transformed to nitrite, which is transformed to nitrate. The only tanks that have zero nitrates are uncycled tanks, tanks with LOTS of plants, or tanks with almost continual water changes. Most of us shoot to keep nitrates under 40 ppm, +/- 20 ppm.
A cycled tank will show zero ammonia and nitrite, and some level of nitrate.
When we talk about 'seeding' a tank, we're simply referring to the process of taking some source of beneficial bacteria from an existing, cycled tank and transferring it into a new tank. Filter media, gravel, squeezings from a sponge filter are all examples of ways to seed (or import bacteria into) a new tank.
Mature media is just filter media that has been in a cycled tank for some period of time (I'd consider it mature after about 3-4 weeks). Using mature media in a new tank is really the same as seeding the tank; you're bringing in the beneficial bacteria on some filter media.
When your tank can process 5 ppm ammonia in 24 hours to zero ammonia and zero nitrite, you can safely assume it's cycled. (To be even safer, see if it can do so for three days in a row.) Then, do a large scale water change - I do 90% or more - to remove as much nitrate as possible. Don't worry about the beneficial bacteria: they're firmly attached to surfaces in the tank and filter. Just be sure to dechlorinate the water as you add it to the tank.
Does that help?
Jim