Serials. Not to be confused with cereals. (Although it does bring up an excellent point: do cereal companies have serials librarians?) Serials is the library term for magazines. Journals. Periodicals. (How many different words do we need to describe the same things?) A serials librarian is thus the person in charge of keeping track of all things serial in the library. (NOTE: yes, I'm simplifying things here a bit, as serials are actually more complex than I'm describing, but I'm trying to keep things more basic.)
So, what does that mean?
Well, it means that they keep watch over all the subscriptions the library has to journals and newspapers. When a journal of newspaper arrives, they process it (stamp it, add a strip so that it'll beep when someone tries to take it out of the library, add a reinforced binding if necessary, and shelve it). They also add it to the catalog holdings. They keep track of what journals are late and what ones never got sent. They throw out old journals if the library only keeps the most recent years. They manage microfilm and microfiche. They might send some unbound periodicals off to get bound, in which case they inspect the bound journals when they return to make sure everything was done right (none are in upside down, none were cut off so you can't read them, etc.).
In addition to this, they get the pleasure of dealing with online journals and databases, managing subscriptions and holdings. If the library has access to a journal online, that has to appear somewhere in the catalog, and it has to be easy to find and understand if a patron comes in to search for it. There are programs and companies that manage this sort of information (Serials Solutions is the one we use), but it still takes work on the library's end to make sure everything's functioning properly.
Serials librarians also keep track of usage statistics. How often are journals are getting used? What's the cost per use of each journal? Should we keep it? Ditch it? These are all areas the serials librarian is expected to at least offer input on, if not make the final decision.
All of these things must be done if the serials collection of a library is going to be easy to use. More complicated than it seems at first, right?