There is no boss fight in this video, just a bunch of awesome scenes, some of which are the best in the game. I thought about cutting out a lot of the in between stuff, but decided against it. 13 minutes isn't terribly long, and you do learn a lot about Kuja in this fight. Also, IMO, this game had the best Bahamut. This was before they turned him into a dumb mecha looking dragon. In fact, FFVII-FFIX all had great Bahamut designs, but this one was my favorite as it looked most realistic.
The only stuff cut out is the walking parts and the few forced fights against the Mist spawn enemies, nothing worth bumping up the time to 16-18 minutes for. Also, there's a small part in the FMV near the end with Bahamut disappearing that normally nearly freezes in the game, so I cut out that tiny part.
Plot
After defeating Soulcage of the Iifa Tree, you return to Madian Sari (Eiko's home). Shortly after some scenes, you find out Eiko is kidnapped by one of the bounty hunters (Lani). Zidane makes a plan to go get her back, but it gets thwarted by her. However, the other bounty hunter knocks Eiko out of her hands as he finds that method of bounty hunting to be disgraceful. This hunter, known as Amarant, appears to know Zidane and battles him 1 vs 1. I skipped that fight, but it's really easy if you're leveled enough and know how to heal with items safely, among other simple tactics.
When the fight ends, Lani escapes and Amarant stops you on the way out back to the Iifa Tree to join your party, sort of unwillingly.
Spoilers for video
Upon entering the Iifa Tree once again, you see Kuja. You find out that he pretty much predicted that the Queen would attack him here, and appears to be fully prepared to deal with it.
Zidane & Company decide they want to stop the Queen from fighting to her doom, but the only Summon Dagger could find, Leviathan, was useless against Kuja, since he's in the air. After some fixed battles against the new Mist spawned enemies, the Queen summons Bahamut to try to take out Kuja, only to have it turn against her and take her out.
And finally, you find out that she was overcome with greed due to Kuja's manipulation of talking her into taking over all the Continents as she comes back to her sense before dying, which now leaves Dagger as the Queen of Alexandria.
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