Can someone please explain how a visitors visa to NZ works for someone with a UK passport? When first entering NZ, the passport is stamped with a 6 month visa. If the person leaves the country, and then returns again within that 6 month period, do they get a new 6 month visa starting from the date they return? Do they have to be away from NZ for a certain amount of time to get a new visa?

Depends on how long you previously stayed in NZ. Remember that as a visitor you are allowed to stay in NZ for a maximum of 9 months in an 18 month period, or 12 in exceptional circumstances (eg you have lodged a PR application)

If you arrived as a visitor, stayed in NZ for 3 months then left (say for a short trip to Oz) your visitor permit expires on leaving the country (regardless that it still has 3 months left on it. When you returned, you would be given another 6 month visitor permit on arrival in NZ which would take you upto the 9 month limit.

If, however, you arrived as a visitor and stayed the whole 6 months, and then left & returned you would only be given a 3 month visitor permit, which again takes you upto 9 months

One thing that often catches people out when doing this is the requirement to have an onward ticket out of NZ. This happened to a good friend of mine who was in NZ on a 6 month visitor permit, who booked a return ticket AKL-SYD-AKL to attend a job interview in Sydney. When he tried to check in for the flight back to NZ, the qantas agent refused to let him check in as he could not show a return or outward ticket OUT of NZ. Only after buying an expensive ticket out of NZ at the airlines ticket desk (which he later managed to cancel & get a refund, minus $50 'admin fee') would they let him board the flight.