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 flair 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).
- "A song of Sun and Shadow": Created by Finmonster, this shared universe has several stories now including Heaven's Light, Seven Days in Corona, The Dragon And The Bow, The Legend of Gobber the Belch, and When The Cold Wind Is A Callin.
- The Great Disney Adventure Saga: A trilogy of fics dealing with a family getting stuck in a Disney Mega Crossover trying to get home.
- The Little Pony Legend: A series of several fics set in a Fusion Fic universe of The Legend Of Korra and My Little Pony: Friendship is Magic.
- Transformers: Avatar Chronicles. Written by The Bane of Humanity, these 5 fics chronicle how the autobots and decepticons got their wars for power and freedom tangled in the world of the Avatar.
- Project Bluefield is one, featuring an Original Generation of OCs who travel between worlds.
- Child of the Storm is the first book of a multi-volume series. To date, it has one complete brief spinoff and one ongoing sequel.
- A Song of Metal and Marvels is a series of A Song of Ice and Fire/Marvel Comics crossovers. As of now, the original story, A Man of Iron is complete, the second book A Crack of Thunder is ongoing, and at least one more book, called A Shield of Man, has been announced.
- Facing the Future Series: A Post-Script Season series of fics chronicling a vision of Danny Phantom and how a 4th season could have gone.
- Land Before Time Retold: While a Re Write of The Land Before Time, it adds its own twists into the story as well. Currently on movie 6.
- The Pony POV Series. Starting with season 2 of My Little Pony: Friendship is Magic, and creating its own vast expanded universe.
- The works of Angelus-v1 in creating his own Young Justice universe. He currently has 4 fics in his universe with several more planned. The ones created thus far: Young Justice: Darkness Falls, Young Justice Titans, Young Justice Thanksgiving, and Young Justice Vow.
- The New Adventures of Invader Zim is currently only composed of one ongoing "season", but Word of God is that it's meant to have several.
- The Elements of Friendship is an ongoing series of re-dramatizing episodes of My Little Pony: Friendship is Magic as epic High Fantasy.