Meep, sorry I missed this question so long ago. Though it looks like many others missed the question too…
The SaGa part is easy. That’s the name of the series. I don’t know if it stands for anything or is just a play on the word saga. As for romancing, definitions 2, 3a and 3b from the link should explain its meaning in the context of Romancing SaGa.
What are the storylines for the Romancing Saga games?
Like the Final Fantasy series, the SaGa series is a collection of unrelated games with numerous recurring themes. Unlike the FF series, the Romancing SaGa series has a wide variety of smaller plots with one general plot arc that ties the game together. I know RS3 is like this, and from what I know of RS1 I think it may be as well. From what I know of RS2, it’s probably a bit more focused on the big plot, but as I haven’t played it I can’t comment.
Anyway, the main plot arc of RS3 is that a phenomenon known as the Death Eclipse occurs every 300 years. The Death Eclipse has the unfortunate effect of killing all newly born life when it occurs. Several centuries ago, one small boy managed to survive the Death Eclipse, and he went on to become a tyrant known as the Devil King. He opened the gates to the Abyss and let terrible monsters roam freely, plunging the world into darkness.
During the next Death Eclipse, another boy lived who went on to become a hero known as the Holy King. With the aid of his allies and some legendary items, he was able to fight off the Abyss Lords and re-seal the gates, bringing a time of peace back to the world. Another Death Eclipse happened a mere 16 years before the game, so it’s just about the time at which the surviving ‘destined child’ will be coming into adulthood. Will he be another Holy King, or another Devil King?
From here you pick up the story based on which character you choose. While each character starts with their own story, and have a few minor twists here and there, the overall game tends to be the same collection of smaller stories as you build towards the ending. The main plot arc is that the Abyss Gates have re-opened enough for the Abyss Lords to send their shadows into the world to stir up trouble. It’s up to whatever hero you choose to seal them, though it seems you start sealing the gates quite by accident. Once you finally seal the fourth Abyss Gate, the identity of the destined child becomes apparent (though you can figure it out a lot sooner). I’ll leave out the last bit of the game, as it’s a big spoiler.
That synopsis is probably way too short and wrong on a few small details, but you should get the gist of it. Of course, that doesn’t help explain the first Romancing SaGa, which is the basis for Minstrel Song. But since I can’t read Japanese and there’s no really good patch out, I haven’t played RS1.