Alignment is the simplest. It takes one vertical line & makes everything either start, end or centre on it. It does nothing else, so the opposite end of each line is free to finish where it needs, without shrinking or stretching and optionally without hyphenation.
This is how anything written on a mechanical typewriter will look*, each line just has to end before you run out of space, no other adjustment is made.
Because left & right alignment are simple 'start from one end' processes, this is as easy to do in physical typesetting with boxes of lead letters as it is for a computer. No 'math' needed. Centre alignment needs measurement, which makes it still simple enough for a computer, but a typesetter would be getting their calculator/slide rule out. Each line is still 'just as long as it needs to be', with the word count adjusted to fit.
To go into Justification, you need two more 'tech words' - Kerning and Tracking.
From Wikipedia -
In typography, kerning is the process of adjusting the spacing between characters in a proportional font, usually to achieve a visually pleasing result. Kerning adjusts the space between individual letterforms, while tracking adjusts spacing uniformly over a range of characters.
Now, as simple as this may sound, kerning is as much art as it is craft. The default gap between letters must be worked out for every potential letter pair. There are some highly complex rules to get the artist/foundry started, but ultimately it comes down to their judgement. A finished font can have between 200 & 1,000 of these dedicated individual pairs mapped.
So, now we come at last to Justification. In it's simplest form, 'fully justified' means that both left & right sides form a perfect vertical line. 'Perfect' however is already subject to human optical perception. Does a letter start from the first vertical, does it start from the beginning of the serif? What about if the line starts with an 'O' which has neither vertical nor serif?
Once that has been decided, then the typesetter/computer must figure out how much extra space to include where necessary to keep the overall look, whilst hitting both edges precisely and not leaving big gaps or crushing letters together.
This is hard.
Dedicated page layout applications such as Quark Express can do it beautifully, time after time. Word processors & such as email apps generally make an ugly mess of it - they tend to prefer to adjust tracking, without keeping an 'eye' on the visual impact kerning is capable of. Sometimes they seem to adjust kerning too… & make an awful salad of it. I have never been able to figure out just how they manage to get it so badly wrong. This is why you should never do pre-press in Word. Any real typesetter will just cry.
I have never used quadding. I've always trusted Quark [or these days often Pages, which is really pretty good at it] to be able to get it right and have never needed [nor probably have the skill] to manually adjust this myself.
I hope this is comprehensible. To simplify it this far I have left huge swathes of technical knowledge and terms out entirely. This is not the full story of how it's done, but I think is potted far enough to not have people nod off halfway through ;)
*OK, so a mechanical typewriter uses a mono-spaced font, but otherwise this is easy to visualise for demonstration purposes.