Does that mean the author announces they're not going to write any more chapters, or does it just mean the author abandoned the fic?

Both.

I wouldn't bother removing it from my favourites or what I'm following. Too much effort. And in cases where the story has been abandoned but the writer hasn't actually said that it'd been ditched, there's always a small chance that he or she could come back to it in future.

True. I've seen authors come back and update so there's always hope :)

I've seen that too. I'd only followed the story in a fit of optimism, because the last update had been years ago, but then the author came back to it.