What do all of these exposure numbers mean to me and my camera? How can I see what settings I would need to take the same shot?
ISO, aperture, and shutter speed are universal standards. A combination of the three represents the correct exposure for a certain light level. If those settings worked for someone, they will work for you in the same light level. You can also play with those settings to match your camera's capabilities.
Here are scales for ISO, aperture, and shutter speed. If you move down the scale on one, you must move up the same amount on one of the other scales. In other words, if you move two steps down on ISO, you must move either two steps up on shutter speed or two steps up on aperture (or one each on shutter speed and aperture) to get the same exposure.
ISO
6400
3200
1600
800
400
200
100
50
Aperture
f/1
f/1.4
f/2
f/2.8
f/4
f/5.6
f/8
f/11
f/16
f/22
f/32
Shutter Speed
1 minute
30 seconds
15 seconds
8 seconds
4 seconds
2 seconds
1 second
1/2 seconds
1/4 seconds
1/8 seconds
1/15 seconds
1/30 seconds
1/60 seconds
1/125 seconds
1/500 seconds
1/1000 seconds
1/2000 seconds
1/4000 seconds
1/8000 seconds
So if you see the settings "1600, f/2.8, 1/60s" and you wonder how that would work at ISO 400, you can do the translation. ISO 400 is two steps down on the ISO chart, so you have to go up two steps on the shutter speed and/or aperture. If you keep the aperture the same, that means that you need to increase the shutter speed from 1/60s to 1/15s.
What if your lens could only go to f/5.6 and you find shots above ISO 800 objectionable on your camera? Then then "ISO 1600, f/2.8, 1/60s" shot would be one step down on the ISO scale (1600 to 800) and two steps down on aperture scale (f/2.8 to f/5.6), so you would need to move up the shutter speed scale three steps to make up for those changes. That would give you a shutter speed of 1/8 seconds.
These rules are the same for everyone. They don't vary based on camera brand or quality. ISO 400 means roughly the same light sensitivity on a camera phone as it does on an $8,000 full frame DSLR. The cameraphone will show A LOT more noise at ISO 400, but the exposure level is the same.