For when you want to see in the dark, know where you are or call a helicopter
A torch is one of those must-have things that you regret not bringing. You use it for seeing in the dark around the campsite or hut, as there is very rarely an unnatural light source at these places. It will also be extremely helpful if you've stuffed up, and so have to walk in the dark. For both of these uses, a head torch is far superior to a hand-torch. Why? Because it allows you to have your hands free, and wherever you look the light will follow. Some people carry both a hand and head torch, which is useful if you're walking in the dark for a long period, but otherwise just a head torch is fine. There is a pretty big range of head-torches available, with $10 supermarket ones at one end and $200+ ones at the other. The really expensive ones usually are very bright, and have funky features like automatically adjusting the brightness depending on how far away you are looking; you will not need one of these. For tramping, you don't really need anything too flash, but don't buy the cheapest you can find as you want something that will last you, which the cheapest ones won't. But if you are just starting out, and not sure if you like tramping yet, a cheap one is fine.
Good features to have in a head torch:
Bright - on the specifications on the side of the box for most head torches you will see listed three important figures, lumens (brightness), range and battery time. If you were prepared to spend a little bit more money (something between $50-$100 on sale), you would be looking for something with at least 100 lumens, at least 70m of claimed range, and at least 50 hours of low light burn time. These figures are all approximate, anything more than this is good if you are prepared to spend some money. But, like I said, if you're just starting out anything is fine.
Red-light mode - this is a highly useful feature to have. The idea is that red-light doesn't disrupt night vision, so it is extremely useful for around the hut or campsite because it doesn't annoy people. In a hut especially, it is considered mildly impolite to shine a bright torch around at night; a red light is perfectly acceptable however.
Waterproof/resistant - it is rare-ish to find a waterproof headtorch, but water resistance is a huge plus. Something at least water resistant to IPX 4 standard (splash proof) is a plus.
varying light modes - the ability to adjust the brightness.
Lock - the ability to 'lock' it so it doesn't turn on in your bag.
Light and compact - you don't want something bulky and heavy.
Uses a convenient battery size - AA or AAA are good.
General Buying Advice:
While it is good to have all of the features above, you don't need much more than that. Really powerful head torches are primary designed for cyclists, you don't need a 1200lumen, 1km range auto -adjusting monstrosity. These will likely be big, heavy and burn through batteries like a hiker burns through chocolate. You want something with good features like the ones above, that is small, light and doesn't break the bank.
If you have a colour choice, it is often a good idea to get a bright colour. This is because a lot of the time when you are looking for it, it will be dark. So although a matt-black head torch may look cool, it will be a lot harder to find in the dark than a bright orange or white one.
Compass + paper maps
These are essential navigation tools. Using them together takes quite a bit of practice, but it is worth the effort. Even if you don't really know how to use a compass beyond figuring out where north is, even that is still hugely useful! So I highly suggest getting a compass and learning how to use it, especially if you are going by yourself or with less experienced people. It is also ALWAYS worth bringing paper maps of where you are going. Not every track is sign-posted, and some are sign-posted misleadingly, so it is always a good idea to have a paper map with you.
As to what you need in a compass, the best ones to use are those with a base-plate (see here for an example, picture coming soon). You can pick up such a compass very cheaply from places like Kathmandu or most outdoor stores. If you are going to be using one a bit more, if you spend $30+ on one it will be a lot more hard-waring. Cheap compasses tend to malfunction very easily. The industry standard compasses are those made by Silva, they can be found in most good outdoor stores.
If you get into something like orienteering or use your compass frequently, I recommend getting a prismatic compass (see picture across), as these make it much easier to get more precise readings.
A watch is another absolutely essential piece of equipment for hiking. The reason is that to do a lot of more advanced navigation you need to be able to time yourself. It is also very useful to know how far away sundown is. Most people nowadays use their phones for the time. However, while hiking you're phone will most likely run out of battery. Your watch doesn't need to be flash, it just needs to tell the time. Anything beyond that is a bonus.
With the increasing prevalence of cell reception, it is more and more useful to bring a cell phone a s an emergency backup. IF there is cell reception, a phone can be a better method of getting help than an emergency locator beacon. Police can now send a text you your phone with a link, which when clicked, allows them to trace your location. Furthermore, they can give you advice on what to do, or how to care for an injured party. That being said, a cell phone is not a replacement for a personal locator beacon, and should be carried in addition to one.
I won't go too far into detail on GPS's here, there is a wealth of information available online about what they do, how they work and the features you can have. On a most basic level, they provide your location, altitude, allow you to track your movements, measure you speed and view maps. However, they are not a replacement for a paper map and compass, but a supplement to them. GPS's are very useful when you can't see (say because of cloud), or have absolutely no idea where you are, and for giving you a location quickly in ideal conditions (open view of the sky). They are also useful for looking at where you have been after the fact. But the reason they are not a replacement for a map and compass is that they can get wet and break or run out of battery; never rely on them exclusively. They are also a bit clumsy for trying to view a map on a large scale, as the screen will often be small and lack detail.
This all being said, they are a useful, but non-essential and expensive piece of gear. I would recommend getting one if you are heading into wild, untracked country where it would be easy to get lost, and aren't sure of your navigational abilities, or just like to have cool stuff.
Personal Locator Beacon (PLB)
These are essentially electronic flares. What they do is send a powerful signal to a satellite system, which is then bounced back to centres around the world, who will send help; in New Zealand that centre is in Wellington. They then pass the information onto local police who will send out a search party or helicopter. They are something of a get-out-of-jail free card for hiking. They are not to be used lightly, but if you are in a perilous situation they may just save your life.
They are very expensive pieces of equipment (~$400-$600), and people usually only make the investment if they get out hiking a lot. Often it is more convenient to hire one, which can usually be done quite cheaply, Many tramping clubs hire them out to their members, and some outdoor stores do so as well.
There are some types which have more features, and allow you to send messages and more than one type of SOS; these are very prevalent and popular in the USA. However they usually rely on a private satellite network, so have less reception in other parts of the world, often work of a subscription basis, and often use a slightly different, less precise method of transmitting a location. Therefor, the best ones for New Zealand are the standard emergency locator beacons available from most outdoor stores. These are limited in their features, but will transmit a very strong signal to any available satellite, and also put out a homing signal to a helicopter which will enable it to find you quicker.
This all being said, when you set one off you may still be waiting 24+ hours, for a number of reasons including the availability of satellite reception, the weather and the availability of a helicopter. Don't just set one off and expect a helicopter in the next hour.