Amateur Radio is the hobby of Radio Communication. Also known as 'Ham Radio'.
Off the top of my head, Recent movies including 'Frequency' (Dennis Quaid) and 'Contact' (Jodie Foster) both made reference to Amateur Radio.
In NZ the deal basically is this:
After taking a basic multi-choice exam covering both Electronics and Radio Theory and Operating Regulations, you obtain an 'Amateur Radio Operators Certificate' approved by the Ministry of Economic Development for operation on the NZ Amateur Radio bands. (Note: Individuals are not licensed. Amateur Radio is now covered by a General User Radio License (GURL) under which those with both an Amateur Radio Operators Certificate of Compentency and a Callsign issued by an approved ARX, can operate).
Once licensed, you take out a Callsign and can begin to use the radio bands allocated to Ham Radio pretty much as you like, within the approved bandplan.
Amateur Radio is popularly regarded as the technology that lets you talk around the world over HF radio equipment similar to that seen in the above referenced movies. Morse Code is also popularised - think 'Independance Day'. The truth is though that there are many dozen different modes and bands available.
The New Zealand Association of Radio Transmitters are the National Association who provide support and administrative services to Ham Radio operators in NZ. They also host information on the AREC and other Amateur services in NZ (including Space/Sattelites, Women in Amateur Radio, the Society for the Protection of Amplitude Modulation and more).