I've never heard of this great seal before, I wasn't taught about it either.
Technically she is the Queen of our country, she has the power to fire our Prime Minister (and has done so once before). But she hardly does anything nowadays, the Prime Minister (currently Julia Gillard) has more impact. We became a country in 1901 and had our first Prime Minister (Sir Edmond Barton) then, so we are still quite a "young" country. If we became a republic and got a President instead not much would change, the Queen probably doesn't have long left to live now, she is getting old...