What is the role of prime ideals? There is a related question regarding this, but almost all answers lean towards algebraic geometry, which I know none of. The wikipedia page says something related to algebraic number theory, another field I know nothing of.
So my question came up during reading Cohen's theorem which states

A ring $R$ is Noetherian iff every prime ideal of $R$ is finitely generated.

Basically this theorem says we can reduce the study to the prime ideals of $R$ instead of just "regular" ideals of $R$ since we know

A ring $R$ is Noetherian iff every ideal of $R$ is finitely generated.

So what is so fundamental in the role prime ideals play? And I don't mean in this particular case, but in general. Do they generate certain structure/objects in algebra like how primes generate all the other integers? Is there something "easier" with regards to prime ideals? Does reducing theorems to prime ideals give us something more fundamental? 
Sorry if these questions sound all the same, I am just looking for some simple answers.
 A: 
So what is so fundamental in the role prime ideals play? And I don't mean in this particular case, but in general. 

This is well answered by the post you linked already. I don't really think "but I don't know anything about those branches so I don't want to hear reasons from those branches" is a valid reason to dismiss the other question. In dismissing them you're dismissing some of the nicest and most relevant cases.
It would be especially worthwhile to read up on the connection between prime ideals and irreducible varieties (for example, something like this).

Do they generate certain structure/objects in algebra like how primes generate all the other integers? 

Well, the direct analogy for that is in Dedekind domains, yes. For those rings, ideals have prime factorizations,  but this isn't true for rings in general.

Is there something "easier" with regards to prime ideals? Does reducing theorems to prime ideals give us something more fundamental?

Yes, that is trivially the case.  If you re-characterize anything from terms of a large class to a smaller class, then you have "made things easier" in that you only have to check the small class instead of the large class.  Reducing questions from the general case to constituents that make up all cases is a general theme.
The example you gave in particular about the relationship to Cohen's theorem was also covered in this earlier post.
