IT'S OKAY YOU AREN'T A PHOTOGRAPHER
At the time I write this, a guest on my tour last night wrote herself off as pretty hopeless when it came to photography. But she had a many, many year old (six year, I believe) point and shoot Sony - before Sony was any good - that she brought with her. It didn't look like much from the outside at all, but there was that little 'M' on the dial at the top, which gave us the ability to manually set the aperture, the ISO to some degree, and to a limited extent, the shutter speed - which was all we needed to achieve a long enough exposure to photograph the night sky.
The excitement in the scream she let out when she saw the photo she had just taken of the aurora flash onto the LCD screen on the back of the camera was something I cannot forget.
Here-in lied a very pure, innocent momentary excitement as well as immediately adding an additional depth to her experience on a frozen lake, underneath the aurora. Perhaps it was as much about having her very own image, she took, of the aurora, as it was a little bit deep down realizing she was capable of more than she initially assumed.
What you need
A DSLR, MIRRORLESS, OR POINT AND SHOOT CAMERA
Any camera that gives you the ability to control, at least a little bit, your aperture, your shutter speed and your ISO. It is worthwhile to explore manual mode on your camera before you arrive to ensure you can individually adjust each of these settings. You are very generally going to want to aim to be able to set your ISO to between 800 and 3200, shutter speed up to 15 seconds (15"), and your aperture to the lowest number you can. f3.5 is okay, f2.8 is great, and anything lower than that is excellent.
Cameras made of metal will conduct the cold a lot faster and more harshly than one made from composite or plastic. As your camera gets colder through the night, you may notice it begin to lag when you change settings or view your images. This is relatively normal.
If you've moved from a DSLR to mirrorless, you've likely noticed the difference between batteries is enough to make you cry. Mirrorless batteries are still not very good even in optimal conditions, and when you take them to extreme temperatures below zero, their performance is really going to suffer.
DSLR batteries are better. I can speak just mostly to Canon here, but DSLR batteries generally are good, even at -30°. You can expect moderately decent performance out of them longer term in the cold. No battery you use will be happy at -40° though.
Batteries on the whole will benefit from being warm. If you can store your spare batteries in an inside pocket next to your body, they will keep better for when you need to switch them out. When a battery dies in the cold, it likely wasn't because it was drained to 0%. Shooting mirrorless, I've had batteries go from 80% straight to dead. But you can get them warmed back up, either inside the car or next to your body, and get more use from them later.
How you shoot will impact your battery life in the cold, just as it does in usual circumstances - but in this environment, the effects will feel multiplied. If you use your LCD screen a lot, you'll notice your batteries drain much quicker than if you're able to adjust your settings by looking through your viewfinder etc. Whatever you do, bring spares.
It doesn't have to be fancy, but you will need one to get clear, sharp images because of the longer exposure time needed.
SHUTTER RELEASE CABLE
Not necessary, but the benefit to having a cable release is a reduced risk of a slightly blurry image from pushing the shutter button and shaking the camera doing so, especially with bulky gloves on. A 2 second timer on your camera is a good alternative if you feel you need it.
Here's the thing. When you bring something that has been outside at -30°C for a few hours, into an environment 50°C warmer, you're going to have condensation develop on the camera. In rare/extreme circumstances on certain cameras, it's possible for condensation to form inside where the mechanics are. While this is something that repeatedly over time may be of concern, you can minimize the condensation issue by placing your camera in a loosely closed ziplock bag when you bring it inside for the night. It will help the camera warm up gradually instead of so instantly.
If you feel that you'll need a light to make adjustments to your camera or tripod while we are out, it's really important that your light or headlamp has a night vision mode - a red light. This is essential to preserve our night vision and minimally impact our other guests sharing this experience. White light is very harsh on our eyes once we have achieved night vision ourselves, and of course the brightness, especially carelessly aimed, can ruin photos as well. Please cover/aim your light consciously.
Before you arrive in Yellowknife and try to figure out how to change your camera's manual settings and adjust your tripod at -30°C in the dark, please practice at home. Just as important here is knowing how to operate your tripod head so you can shift your camera's view effortlessly.
When we are out in the countryside, with no real lights in the distance, and no moon, your autofocus will become useless (because there is nothing of significant contrast for it to lock on to). This means you're going to need to know how to manually focus your lens to infinity (so the stars and aurora are sharp/in focus), which is going to be near that little infinity loop - ∞ - on your lens, but not always necessarily right in the middle of it.
For this, there are a few things you can do - from your accommodation in Yellowknife, use your autofocus to lock onto some very distant city lights (or landscape if it is still bright out), switch your lens/camera back to MF - manual focus - and note where your lens distance scale markings are aligned, or even better, scotch tape your focus ring to the barrel of the lens so you cannot accidentally move the focus ring. This will help you achieve (near) infinity focus for the night. You may also be able to use your camera's live view function to visually adjust your focus until you can see that the stars showing on your LCD screen are sharp.
Bring a microfibre cloth. In the fall aurora season, we can often have very humid nights near the lakes which means your lens (and camera) will become coated with condensation. In the winter, ice crystals can form on your camera and lens, and obviously you'll want to clear your lens in both these situations.
Remove any filters (including UV) from your lens when shooting at night.
iPHONES & MOBILE PHONES
Android users - you, for the most part, are able to control your camera's advanced settings - like shutter speed and ISO. This is very good news. You will want your ISO set as high as possible, and a longer shutter speed of a few seconds selected.
iPhone users - you, absolutely, are not able to control your camera's advanced settings, which means for the most part, you won't be able to capture much.
Thankfully, there are app developers who have our back.
iPhone 7 & older: My favourite when I was using my iPhone 6 was NightCap Camera. This will give you full control over your ISO, focus, and shutter speed. There is a learning curve, but it's a good option for a couple dollars. Northern Lights Photo Taker is a more simple option with a point and shoot feel and overall lower quality images, but will still increase your iPhone's ability to capture a little bit of the northern lights.
iPhone 8 & iPhone X: In the release of the newest iPhones, the camera changed substantially. Apple also introduced a new noise reduction system that can make the images feel of a lesser quality in the end, and almost kind of like they've been highly 'compressed'. There are ways around this, like with NightCap to use the 'stars mode', but you'll need a tripod or to steady your camera. The folks at NightCap have also written a guide to help you photograph the aurora with their app which is worth reading.
I've searched high and low and spent $3 way too many times trying almost every camera app out there. Camera+ and ProCam 5 are maybe two of the 'best' so far, but the results may not exactly exhilarate you. Camera+ you can work handheld (with a lot of grain in your photos), and ProCam 5 will be better if you are able to use a tripod to steady your phone for longer exposure times.