I think it's fair to say that almost every character in FFVI has a sad and/or deeply touching moment. From Terra learning of her past, and how to love with Leo and the kids, to Cyan seeing everything he loved destroyed and trying to cope with loneliness. I even found Edgar rigging the coin flip to sacrifice his own happiness for his brother's sake to be a somewhat emotional moment.

I also would've liked to have seen a little more development for Mog. When you find him in the World of Ruin, all of the other Moogles have vanished without a trace and it would've been good to see how he dealt with being the last Moogle. (if indeed he was)

That's just the tip of the iceberg - I haven't even mentioned the likes of Celes, Locke or Shadow, all of whom had some really sad stories. Whose story touched you the most?