Australian English began diverging from British English shortly after the founding of the colony of New South Wales in 1788. It arose from the intermingling of early settlers from a great variety of mutually intelligible dialectal regions of the British Isles and has, over the following centuries developed and diverged into a distinct and unique major variety of English—though still keeping British English spellings to words like colour, civilisation and metre, thank-you-very-much.
Records from the early 19th century survive to this day describing the distinct dialect that had surfaced in the colonies since first settlement in 1788, with Peter Miller Cunningham's 1827 book Two Years in New South Wales, describing the distinctive accent and vocabulary of the native born colonists, different to that of their parents and with a strong Cockney influence.
Since World War II, American media has made a profound impact on the language—particuarly slang. However, it's still more common to use Aussie-isms, as using words overly-typically American or British sounding will earn you ridicule.
For examples of modern Aussie uniqueness and peculiar words, have a gander at Australian Accents.