Trying to answer in regards to the specific definition of the roles and how they came to be, not specific to StackExchange.
An administrator has authority over a system that a regular person does not. They have elevated access. You can compare this e.g. to the police, who are able to avoid certain parts of the law when they are performing their duties (e.g. speeding to catch a criminal).
The elevated rights of an administrator are usually potentially damaging if the admin intends malice, but the administrators are expected to only wield their power for "benevolent purposes".
The consequence of this is that the normal users rely on the administrator for certain tasks. This is by design. The elevated rights are only given to a subset of users (the admins), and the admins are then expected to help the users with their problems.
As an analogy: if we give a police officer the right to speed when in pursuit of a criminal, that means we expect them to try and catch criminals.
But here's a problem: the administrators often get overloaded with irrelevant questions. Since a normal user does not have the power to e.g. ban other users; they also don't know the criteria that warrant a user being banned. User A might flag user B for banning because he always makes typos, or he disagrees with user A.
As a result, administrators spend more of their time telling people that they can't help them with what they're asking.
To continue the analogy: the police have to explain to a civilian that they cannot just "steal back" the item that someone stole from the civilian.
Enter the role of a moderator. Moderators are the initial point of contact. They listen to the complaints of a user, check if it's something that the admins are able to help with (and allowed to), and only then send the work on to an administrator.
To continue the analogy: imagine if we made a police helpline. Not intended to report emergencies, but to tell people whether their complaint is valid enough that they can file a police report.
By doing this, the administrators spend most of their time only working on valid complaints, instead of continually having to explain to people why they cannot help them.
Edit I want to add that moderators can sometime be "assistant administrators". I.e. a moderator can be given certain rights to solve simpler problems, whereas the administrator is only expected to handle the more complex problems that moderators cannot solve by themselves.
Helpdesks and callcentres often use a multi-tiered approach:
- First line takes the calls and talks to the user (= moderator)
- Second line handles slightly more complex issues that first line cannot. They are the "expert" first line (= super moderator?).
- Third line are resolver groups. They handle large scale issues (e.g. a mail server outage), because first/second line only handle problems that pertain to a single user (e.g. Outlook is not installed on the laptop) (= administrator)
- Fourth line are the architects. They do not help with solving technical problems that occur; they are focused on creating a better system where less problems will occur. This is where we move from IT support (the first 3 tiers) to software development (tier 4 and up). (= super administrator)
Second edit A small analogy to show that this is not limited to the IT sector:
- If you are hurt, an EMT will come to your aid. If he cannot fix your problem, he will take you to a hospital.
- At the hospital, a nurse will look at your case, and see if your problem requires further medical attention.
- If it does, then the nurse calls in the doctor. The doctor assesses the required treatment.
- If it involves surgery, the doctor then calls in the surgeon.
Same tier system, and it exists for the same reason. No surgeon wants to be disturbed just so he can look at a mild case of the flu.