If you have applied for New Zealand residence but the application has been declined by Immigration New Zealand, you will have a right of appeal within 42 days of the date of decision under the Immigration Act 2009.
Appeals lodged will be heard by the Immigration and Protection Tribunal which is part of the Ministry of Justice. You will need to show how Immigration New Zealand's decision to decline your application was incorrect, and/or whether there are exceptional circumstances which warrant the grant of residence visa regardless. You will need to make references to the relevant immigration instructions and past Immigration and Protection Tribunal decisions back up your case.
While there is no requirement that you must be represented by a lawyer, for best chances of success, it would be advisable to engage a suitably experienced legal professional to undertake the work on your behalf.