Depending on your area of study, you may be able to stay and work in New Zealand for up to four years after finishing your studies, and possibly gain residence. However, you will need the appropriate visa to do this.
International students who have achieved a New Zealand qualification can apply for a Post-study work visa or Post-study work visa (employer assisted). These allow you to stay in New Zealand while you look for a job and gain work experience in your field.
This visa gives you up to 12 months to find a job in a field related to your studies. While you are looking for a job in your field, you are allowed to work in any job to support yourself.