If you or your child was adopted in an overseas court, or through an administrative process overseas (ie not through the New Zealand courts), he or she will probably have gained New Zealand Citizenship by Descent. NZ Citizens by Descent have all the rights of New Zealand citizenship, except that they cannot pass on their citizenship to any children born to them (or adopted) overseas.
This can create huge problems if you (the adopted person) go to live in Australia or elsewhere. See this TVNZ news clip to understand the issue. People born in Australia or other countries to NZ citizens face a similar situation. http://tvnz.co.nz/seven-sharp/citizen-shame-video-5853597
The adopted person needs to apply for Citizenship by Grant, after he/she has been living in New Zealand for a year or more. This citizenship requires the adopted child/adult to be living in NZ. The application can be done by the adoptive parent, or by the adopted person themself once they are 16. See information on changing your status from Citizenship by Descent to Citizenship by Grant on the Dept of Internal Affairs website.
Normally anyone applying for Citizenship by Grant must have resided in NZ for at least five years, but for children, if the adoptive parents have lived in New Zealand for more than five years, you can apply for your child's NZ Citizenship well before 5 years, even immediately after you return to NZ with your child in some cases.
You can download the form for applying for Citizenship by Grant at the New Zealand Department of Internal Affairs website. Internal Affairs have advised ICANZ than you don't need to resubmit the original adoption documents as these have already been sighted and recorded by INternal Affairs.
Processes, costs, requirements and DIA staff do change, so if in doubt call the Wellington Citizenship office - NOT your local DIA office. Your adoption documents are vital, often irreplaceable documents - enclose a self addressed return courier envelope and record the tracking number of both your courier envelopes. Make photocopies of everything before you send them away and enclose a list to make it clear what you have sent. Keep originals in a VERY safe place.