Blatantly stealing from my own answer on Graphic Design SE, but accounting for Academia’s peculiarities:
First of all, the following are “duties” for the moderators as a team.
An individual moderator can always decline a specific task.
Also, a moderator can always take a break when they see fit.
I would group the main moderator activities into three categories:
This can be all sorts of things from rudeness over migration requests to too many comments on a post. You can find some statistics on flag handling here, but it doesn’t tell you what kind of flags we get.
Most importantly, the amount of time consumed by different flags varies wildly. For example, a clear case of a question posted as an answer with an existing constructive comment can be handled in five seconds. However, a too-many-comments flag usually requires you to read the question, answer, and all the comments and then make a decision which should be left where they are, completely deleted, or moved to a chatroom.
The most prominent kind of flags in times of consumed handling time are certainly too-many-comments and individual comment flags. Since Academia by nature attracts a lot of comments, it also generates a lot of these flags.
Investigating and addressing problematic behaviour
This can be users who are frequently rude, commit vote fraud, perform consistently bad reviews, post subtle spam, vandalise their own posts, etc. Moderators have the tools to investigate this and react accordingly, which usually involves at least a moderator message and often a suspension.
Such cases are more time-consuming than the average flag, but they happen less often. In 2018, we sent 48 mod messages, and this probably counts multiple messages, when there was some back and forth between the user and the moderators.
The challenge here is to gather the relevant information (with tools), read and assess it, decide how to react, and finally to communicate the result to the user in a constructive manner.
I second StrongBad’s answer that one of the perks of Academia SE is there are very few users in the grey zone where you don’t know whether to do nothing, to warn, or to suspend.
On the other hand, since we deal with life-changing decisions, sensitive information, and ideologically charged topics quite frequently, things occasionally get serious.
Typical manifestations are suicidal users, users wanting to purge all their content from the site, and users calling you a nazi because you deny them their constitutional right to be an arsehole.
Moderators can change some parts of the site’s interface and mechanics such as certain parts of the help centre, close reasons, tag synonyms, etc. In most cases, these are initiated by a community consensus. Moderators also act as an interface between the community and Stack Exchange (the company).
In my opinion, moderators also need to incite discussions on meta and propose community rules if there is a pressing need.
Sometimes, only moderators can notice such problems since they have access to more information and see the community from a different angle.
However, on Academia often the community takes the initiative here, and there are few such issues nowadays, since – at least as I see it – our policies have converged to something that works well and finds broad agreement within the community.
These activities can consume some time, but if they do, they are fortunately not very urgent.