Like most of the rest of the world (excluding USA), New Zealand has the GSM cellular system. If your cell phone does not work on this system, you may buy a phone in New Zealand or obtain a pre-paid SIM card or plan from a service provider.
Mobile phone coverage in New Zealand is very good with all towns and cities receiving mobile phone coverage from the two main players. However if you plan to visit the countryside, you must be aware that there are large areas with no coverage: in many National Parks you are unlikely to get coverage.
Vodafone is an accepted provider in New Zealand, along with XT Mobile Network (Telecom New Zealand) but you might need to buy a specific phone for using this provider. Vodafone can supply you with a pre-paid sim card for your phone on a pre pay or Pay As You Go solution.