First, the dates
Two of them.
The date you were notified of the decision. Appeal rights run from this, not from the day you read it, and they are counted in days. For a residence class decision the Immigration and Protection Tribunal must receive your appeal, and the fee, within 42 days of that notification. If the decision was posted or couriered to you, the 42 days start from the date it was delivered to your contact address, which is not always the date printed on the letter. The 42 days include weekends, though not a public holiday falling on a weekday or your regional anniversary day.
Your current visa’s expiry. If the declined application was going to keep you lawful, this is now the more urgent date. Staying past it makes you unlawfully in New Zealand, which changes what you can apply for and creates deportation liability.
Work both out before doing anything else.
What was actually decided
The wording distinguishes between things that look similar and are not.
Not satisfied that a requirement was met. The officer did not have enough to be sure. Often a documentation problem. This is the most recoverable position, and a fresh application with the right evidence can succeed.
A requirement was not met. A definite finding. A fresh application only helps if the underlying fact has genuinely changed, for example the role now pays the required rate.
Character or credibility. The officer found that information was false or misleading, or that something was not disclosed. This is the serious category. It follows you into every future application, including to other countries that ask whether you have been refused a visa. It has to be addressed directly rather than worked around.
A discretion was exercised against you. The officer had a choice and made it. Whether that can be revisited depends on the visa type.
If the letter is not clear to you, have it read by someone licensed before you spend money on the next step. The category determines everything that follows.
Which options exist
Residence class visas can generally be appealed to the Immigration and Protection Tribunal, which is part of the Ministry of Justice rather than Immigration New Zealand. The Tribunal must receive the appeal and the fee within 42 days of the date you were notified of the decision. Missing it normally ends the option entirely.
Temporary visas usually cannot be appealed in the same way. In practice the routes are a fresh application that addresses the reason, or a request to have the decision reconsidered where that is available.
If you are already unlawful, a request under section 61 of the Immigration Act is a separate route. It is entirely discretionary, there is no appeal from it, and the longer the situation runs the harder it becomes.
When reapplying is a mistake
Reapplying works when the problem was a missing document and you now have it, or when a fact has genuinely changed.
It fails, expensively, when nothing has changed except the hope of a different officer. A second decline on the same grounds is harder to recover from than the first, because it establishes a pattern, and each refusal is disclosable in future applications here and elsewhere.
The question to answer before paying another fee is not “what did I leave out” but “what was the officer not satisfied about, and is that different now”.
What not to do
Do not go quiet. Deadlines pass and options close on their own.
Do not leave New Zealand without advice. Being offshore changes which rules apply and can remove options that were available.
Do not submit a fresh application purely to buy time. If the underlying problem is unresolved, you are usually paying to be declined again.
Do not take a paid job offer as the fix. If the decline was about character or credibility, no employer can solve it.
Getting help
Only a licensed immigration adviser, a New Zealand lawyer, or a person with a specific exemption may give you immigration advice. After a decline the options narrow with time, so the cheapest point to get that advice is the week the letter arrives.