This is the single most common mix-up on this topic, and it is worth getting right before you book. Australian citizenship and Australian permanent residence are treated differently by New Zealand.
If you are an Australian citizen
You do not need a visa and you do not need an NZeTA. Immigration New Zealand lists Australian citizens among the travellers who need neither. You still travel on your Australian passport and you still complete the traveller declaration like everyone else.
If you are an Australian permanent resident
You need an NZeTA. Permanent residence in Australia does not carry the same entry right as citizenship, and airlines check for the authority before they let you board. Apply before you travel, not at the airport.
Holding an Australian visa is not the same as holding Australian permanent residence. If you are unsure which you hold, check the grant letter from the Department of Home Affairs before you rely on it.
Travelling on to Australia afterwards
If you pass through New Zealand on your way to or from Australia, you need a current visa from the Australian government to enter Australia. New Zealand does not issue that, and an NZeTA does not stand in for it.
Source: Immigration New Zealand — New Zealand Electronic Travel Authority (NZeTA)
More on the NZeTA
What is the NZeTA
The NZeTA is a pre-travel authorisation for visa-waiver passports, not a visa. What it covers, who needs one, and what it does not do.
How to apply
Applying takes about ten minutes in the official app (NZ$17) or on the Immigration New Zealand website (NZ$23). Here is the order of the steps and what trips people up.
Cost
The NZeTA request fee is NZ$17 in the app or NZ$23 online, plus the NZ$100 IVL — about NZ$117–NZ$123 in total.
What you need
A valid passport, a photo of your face, an email address and a payment card. What the photo has to look like and what else is asked.
Processing time
Most NZeTA requests are decided in minutes, but Immigration New Zealand asks you to allow up to 72 hours. What makes an application take longer.
Validity
An NZeTA lasts 2 years from the date it is issued, or until your passport expires — whichever comes first — and allows multiple entries.