As a rule of thumb, moderators can do things to posts and threads, while administrators can do things to posters and forums.
For the day-to-day work, the moderators keep an eye on their forums, both from reports and their own reading. The very first step to becoming a moderator, of course, is being an active member. You also develop an instinct for which threads are likely see problems cropping up-- It took me years to break the habit of opening up suspect threads I wasn’t interested in, just to see if there was a problem.
When they see a problem, they take action. Sometimes it’s simple: You see blatant spam, you nuke it, along with the spammer. You see a thread starting to get uncivil, you put in a mod note, and maybe, in extreme cases, a warning or close the thread. Someone accidentally put a thread in the wrong forum, or it’s drifted into another forum’s territory, you move it. All of this, a moderator can do by themselves.
But then there are situations where one moderator alone can’t handle it. In these cases, they don’t go up the hierarchy, like you might think. Or rather, they do, but they don’t just go up: They go sideways. The mod that first noticed the problem e-mails all of the other mods, and they discuss the situation behind the scenes and come to an agreement about what to do. I think warnings also go into this moderator loop, even though a mod can issue them solo (this has gotten a bit more formal since my day). This is the sort of process you get before a banning, or the like.
There’s also a behind-the-scenes forum on the board, only visible to moderators, that is used for documenting and sharing information. I don’t know all the details of this, though, since it was after my time.
