Birds Australia has a long history. It started in the 19thC and for most of it's life was known as the Royal Australasian Onothological Union. They publish "Emu" a scientific journal about birds, "HANZAB" a truely monumental 6 volume publication that is the standard reference for birds in Australia and also the Australian Bird Atlas.
They have branches in every state and some regions, are headquarted in Melbourne, and also run three? observatories and two? large reserves. They are as has been said prime movers in bird conservation and run the Threatened Bird Network.
The Discovery Centre is something new, and is set in an old navy base. I don't think it is even open to the public on a regular basis yet. No captive birds though, so it is outside the scope of this forum.
Interestingly it is about to merge with the Bird Observation Conservation Australia, formerally the Bird Observers Club of Australia. Almost as old, this organisation dates back to 1905 and is also based in Melbourne. It is probably a little more grass roots, as it has many more local branches than BA.
The new organisation will be known as Birdlife Australia, which no doubt is derived from Birdlife International, the international co-ordinating organisation for bird conservation.
Hope that helps.