Along with the form which you have to fill in,
http://www.immigration.govt.nz/NR/rd.../0/INZ1146.pdf, you need to look at the relevant guide,
http://www.immigration.govt.nz/NR/rd.../0/INZ1199.pdf, which also refers to the form your partner needs to use to apply.
http://www.immigration.govt.nz/NR/rd.../0/INZ1198.pdf
Your partner and you are going to need to prove your partnership, for which one lot of evidence is necessary, then you are going to need to prove your visa and financial status, for which another lot of evidence is necessary.