Fanfiction authors do several stories, but they don't have any continuity with one another. For instance, the hero may be in a different situation in story B than they were in story A, while story C will be an AU whereas the first two were following the canon straight.
But then there are some that decide that they want to create more than just a one-shot, or a multi-chapter tale. They instead decide to give their stories a canon of their own, and, as such, each of their stories can pick up where the last left off. They could make a sequel to their own story, or they could begin a series fic (can also be called a serial fic due to the fic acting like a serial series on television).
In a series fic, each story follows a certain continuity with the one that preceded it. In fandoms, these can be tricky to pull off well due to the reader having to sift through several stories to catch up if the series runs on for long enough. Furthermore, the author needs to convey to the reader that not all plot lines are designed to conclude at the end of the story (in the case where a series is planned before the first story is made). Authors also must make aware that a particular fic is part of that series continuity, especially if the canon characters have become out of character by the time the story chronologically takes place that the reader might be put off if they don't realize that it is deep into it's own alternate timeline.
However, if done well, they can be seen as epic pieces, and could be a crowning moment for the author. Keeping someone engaged for as long as a series might last (or, as long as the author is willing to continue it) can be a feat in of itself, but to keep a sense of who the canon characters are while putting new flare to them for as long as it holds out is even more daring. The rewards are great, but the risk is just as large.
Series fics can be rather fun to write, as well, because authors can choose to leave the last chapter in a particular story in the series with a cliffhanger
that can resolve itself in the first chapter of the next story in the series. They can also have some interesting situations that the canon characters can get into as a direct result of the story lines arching over multiple stories, and a cast of original characters that the author can be as creative as they want with (within reason, of course). As said before, greater risk can lead to a greater reward.
Most authors will, in this sense, make summary chapters in fics that are included in these series so the reader can get caught up without having to reread dozens of chapters of the previous fics to have the same effect. They can also direct them to places where they can get caught up (such as a website devoted to their series fic) if they are more comfortable with that method.
Series fics can begin at mainly any point in the source canon. However, once it starts, the series can steer off in its own direction with the stories and with its characters. Because of the nature of series fics, more tolerance of original characters, and canon character going out of character after a certain amount of the series has been played out, are given if it's an ongoing series with no set end point.
Compare Alternate Universe Fic
(which these can loosly be called), Continuation
(those can become series fics in their own right), and Original Flavour
(which many series fics are known, to a lesser extent, to have as a rule).