What we're supposed to believe:
Syrio Forel, the former First Sword of Braavos, becomes Arya's swordfighting "dancing" instructor and dies offscreen while defending her from Lannister guards following Eddard Stark's arrest. He must be dead, because Ser Meryn Trant, one of his adversaries, continues to appear in the series, and that dude seems fine.
The theory:
Syrio Forel is actually one of the Faceless Men from Braavos -- i.e., an assassin who can change his appearance. If that's the case, it wouldn't be much of a stretch to suppose that Forel is also Jaqen H'ghar, the Faceless man who killed three people for Arya. It makes sense. As this Dorkly video points out, both Forel and H'ghar are from Braavos, they both offer Arya their help and they both call her "boy.