I am now a dual citizen of USA/AUS. Born Australia. Work, live, married, run my own small USA business, pay USA taxes, resident in USA 20 years. In the 20 years I've had no income from work in Australia, no Australian tax return. A client just asked me to give one week's project work in Sydney, earning approx ($30,000AUS) paid from Sydney accounting. Do I roll that into my USA tax only? Or do I have to pay Australian tax and not USA? Or both?
As I understand it, Australia is like most jurisdictions in that it primarily taxes by residence, but, also like most jurisdictions, they tax any income earned in Australia by nonresidents. So you'd have to pay tax on the AUD 30,000 to Australia. A good place to start is probably the ATO's page Work out your residency status for tax purposes.
You must also pay tax to the US on the AUD 30,000, because the US taxes its citizens (among other classes of people) on their worldwide income, no matter where they reside. You can normally take a credit for tax you pay to Australia, although you can also deduct it. It is usually more advantageous to take the credit. See the US IRS page on Foreign Tax Credit.