To perform a bushman's blow is to press one nostril closed with a finger (or more traditionally and less cautiously, a thumb), turn one's head to the side, and vigorously blow the contents out onto the ground and (most likely) one's face. Also known (thanks tWD) as a "Bushman's Hanky".
The bushman's is named for the fact that your average "bushman" in Australia does not carry around luxuries like tissues or handkerchiefs, and as such has to blow his nose in the only way available to him. Many Australian men will be very impressed at anyone who can execute this without covering their chin, cheeks and mouth with snot. They will cry "FERAL!" and "GROUSE!" and other such nonsensical expressions of awe. All Australian women (all those worth mentioning at least) will find this disgusting, and will most certainly NOT have sex with you if they see you do it unless you do something suitably impressive afterwards, like a jug scull, or perhaps a quality burnout. In particularly low-class areas a roll-back will suffice, but you will be thought of as a "wuss" with a "sheila's car" if you fail.
The ability to perform a good "bushman's" is mostly innate. If you have particularly protruding lips or chin you have no chance, and obviously nose shape comes into it a lot. Anyone with any kind of facial hair beyond sideburns is obviously out of the game before they even start. The type of blockage you're trying to clear also plays a big part. If you have a runny nose and try a bushie, you'll hose your shirt, face, and possibly people in the next state. If you have a large 'nugget' of solidified snot in your nasal protruberance, you're in with a good fighting chance of getting away clean.
If you do attempt a "bushman's" and fail, the best way to deal with it is to immediately wipe your face with your hand (I know, it's icky, but so is having a face covered in snot), and find a rag, a tissue, or some water as soon as possible to clean your hand and face. If none of these are available, you're probably genuinely out in the bush, in which case wiping on your clothes should be perfectly socially acceptable. This IS a nation of convicts after all - except for Western Australia.
I'm advised that in the USA this act is known as either a warehouse blow (thanks dannye) or a snot rocket (thanks drenehtsral). Badass.