These are my targets. Most have been mentioned already but I’ll summarize:
Spiders - be sure to check both ground and vegetation above. I wear a head band with various levels of illumination (can also help with the photos when not using my camera flash).
Harvestmen (Opilionoids)
Earthworms; and any other worm-type; and Centipedes; Millipedes;
Flies: there are some that seem more nocturnal (of course mosquitoes and I sometimes take a picture of those)
things that look like flies but are not (ha ha). There are some cool “others” out there at night.
Beetles: lots to find…
Orthopterans, esp katydids and crickets.
Moths of course; also caterpillars, esp if you carry a portable black light
Some mammals: skunk, fox (2 species), southern flying squirrel, white-tailed deer, oppossum, raccoon…
Frogs, salamanders, toads
Although I’m a bird person by day, I don’t pay much attention at night although I do enjoy the sounds of any owl.
I’m sure I’ve missed one or more but it’s time for bed… 
I’m in the Southeastern US.
I use a “bridge” camera (Panasonic Lumix FZ1000; I used to use a Nikon or Canon point and shoot). I will use the flash on my camera, but note that it does change the color of the image; but the clarity is really nice. I will also use my Fenix headlamp which can be set to a very high output, but it’s a lot more awkward when I’m by myself (b/c I have to hold the flashlight to the side of the camera to get the best light coverage and that can cause me issues). In some situations, only the camera flash will suffice.
I try to remember to make a note about a given image being “flashed” when I upload to iNaturalist.