There are many different types of astrophotography. Depending on what type you want to do, you'll need (and won't need) various parts and accessories.
Nightscapes
Put very simply, nightscapes are just landscape photos taken at night. There are Milky Way nightscapes, Orion nightscapes, Star Trail nightscapes, etc.
1) Camera - Any camera (preferably DSLR or Mirrorless) that allows you to manually control it's Shutter Speed, Aperture and ISO will do. These settings need to be manually tuned each session, which cameras won't automatically do on their own. (Shutter speed needs to be as long as possible before stars appear to stretch due to Earth's rotation, aperture needs to be all the way open, and ISO needs to be as low as possible to minimize undesirable noise in the picture. (Also make sure it can shoot in RAW file format for more editing power when you get the files home) Lastly, it's preferred if you can delay the shutter (or remotely activate it) to avoid shaking the camera as you take the picture with your hand.
Full-Frame cameras have larger sensors and are therefore more expensive.
Crop-Sensor cameras have cropped sensors and cropped prices. Crop factors need to be considered and applied when picking/using different focal lengths.
I personally started with the Canon T7i, then upgraded to the Sony A7iii (for nightscapes) when I decided to fully dive in. Now I'm testing the Canon T7i with a small refractor for (wide) deep space.
2) Camera lens - the wider (shorter focal length) and faster (lower f number) the better. I have always used lenses that are 20-28mm in focal length, and no 'slower' than f3.5 (For reference, my favorite wide/nightscape lens is 20mm, with a very 'fast' f1.8 aperture)
Note - practice manually focusing it during it the day, so you're ready to focus the lens on the tiny stars at night. (autofocus doesn't do great at all) This may require switching not only the camera to 'Manual focus' mode, but the lens as well depending on your camera type.
3) Tripod - A steady tripod that won't sway or vibrate in the environment. Some people get anti-vibration pads to place the tripod feet on, others hang a bag or weight of some sort from a tripod hook to weigh it down. (be mindful of wind/sway)
4) Star tracker - A "Star Tracker" is a small camera mount that goes in between the camera and the tripod. These devices spin in the opposite direction of the Earth (once you've polar-aligned them) and allow you to take longer star photos before any apparent star motion. Without this, you'll notice stars stretching after just seconds with any lens. I'm happy I learned with the Move Shoot Move, the whole concept of polar aligning was practiced very easily with it's bundled laser.
- Two bonus facts here: Star trackers AREN'T completely necessary if you just stick to very short shutter speeds and 'stack' the sub-exposures into a master exposure file. Secondly, star trackers DON'T track stars, despite what anyone has told you. All the real star tracking is done in a Deep Sky Astrophotography process called 'autoguiding.' (which I'll cover below, #yourewelcome )
Deep Space
Every type of astrophotography done through a telescope is generally regarded as Deep Space/Deep Sky, no matter how wide or narrow the telescope Field of View is.
Before buying a telescope, it's very important to first pick your ideal targets. Do you primarily want to photograph objects within the solar system? Distant Galaxies? Nebulae? What are the angular sizes of these objects and what telescope/camera combinations will give me the perfect Field of View?
1) Telescope OTA (Optical Tube Assembly) - very generally, you'll see refractor telescopes used for nebulae, SCT telescopes used for distant galaxies, and reflectors for planetary or visual observation.
Telescope aperture is the diameter of it's face. (as opposed to camera lens aperture) The larger the telescope aperture, the more surface it can collect light through. Reflectors/dobsonians can provide huge apertures for relatively cheap since they use mirrors instead of lenses like refractors.
Refractors use lenses (two in doublet refractors, three in triplet refractors) to hone the light in from the front of the telescope to the back. (I recommend triplet refractors with 'ED' glass for the nearest/largest galaxies and nebulae)
Reflectors use mirrors to bounce incoming light to the eyepiece/camera. (I recommend these for visual observation, not astrophotography)
SCT's use a combination of lens/mirrors and have incredible focal lengths in compact packages. (I recommend these for the most distant/smallest galaxies and nebulae)
2) Reducer/Flattener/Coma Corrector - As if everything going on inside the OTA wasn't complicated enough, there's still more work that has to be done for the best final product. Depending on the type of telescope, you'll benefit from these to correct for inherent aberrations/imperfections. Reducers are called so due to their focal-length-reducing effect. A 'fast' telescope has a good aperture/focal length ratio, so reducing the focal length while keeping the aperture the same will yield a more powerful light-gathering system.
3) Astronomy Camera (Cooled preferred) - you can absolutely attach your DSLR/Mirrorless camera to your telescope (with accessories: t-ring and t-adapter) but you'll inevitably fight thermal noise as you heat up the sensors with longer exposure time. With dedicated astronomy cameras, you'll see cooling features that tackle this problem from the very beginning.
These astronomy cameras also come in 'monochrome' versions. With specific filter combinations, you acquire a series of highly detailed (black and white) photos for color-merging afterwards. (very advanced)
Note - cameras need to be carefully positioned behind the telescope with enough space for proper 'backfocus.' Without paying attention to this spacing, you'll never be able to focus your camera.
4) Autoguider: (GuideScope and GuideCamera) - yes.. you need a telescope for your telescope to take pictures while you take pictures. The job of the piggy-backed autoguider is to actually track stars. (unlike a 'star tracker' previously mentioned) This mini telescope will monitor and send corrections to your mount, because even the most expensive mounts are just mechanical objects. (designed solely to counter Earth's rotation as smooth as possible.
These products were always controlled with computer software, which meant you had to haul your laptop around with all this equipment as well. Nowadays, with the right products (Such as the ASIAIR Plus), it's all controlled through your phone or tablet. Incredible right?
5) Equatorial Mount and Tripod - essentially, these are like the star trackers mentioned earlier for cameras, but EQ mounts are for telescopes. Be careful to avoid 'Alt/Az' mounts though, as these do not smoothly counter the spin of the earth in the way that astrophotography demands. (Unless you plan to modify the Alt/Az with an EQ 'wedge.'
Note: Most electronic mounts these days have 'Go-To' computers in them. Once oriented on the Earth (polar-aligned) these amazing products will automatically point your telescope onto any target in the night sky. Just scroll through the list of choices, and hit Go-To.
6) Large Portable Battery - if you're away from home, under dark skies, you'll need a reliable way to keep everything powered on. Make sure to protect it from the elements overnight; I keep a GoLabs R300 in an insulated lunchbox and it keeps everything powered with battery to spare after a full night of use.
Conclusion
As far as equipment goes, this pretty much sums it up. Everything can be further complicated plenty, but I'll be covering more in future posts. This is for everyone wishing there was just a straight-forward overview somewhere without having to search all over different sites and forums.
Don't forget to mind the light pollution, be safe, and clear skies.