The final boss is the hardest (duh) you will ever face in the game.  First off, you need a LOT of healing items and some good attack items.  I recommend using Tidus, Wakka, and Auron.  Also, I know there are a lot of things on this page that talk about the final boss, but none of them really go into full detail.  So, here's your copy of "Slaying Sin For Dummies"
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Braska's Final Aeon-Phase 1
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After the scene with Jecht, you must face Braska's Final Aeon.  He really isn't too hard, but he's still a pain.  He only has 60,000 HP.  However, watch out for his overdrive "Triumphant Grasp" because it'll Zombie you if you don't die from it.  Also be careful of his Jecht Beam, because that can (and usually will) petrify you.  To get through this, have Tidus use Haste or Hastega on all your party members.  Use Tidus and Auron to attack Braska's Final Aeon.
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Yu Pagodas
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The biggest pest problem lies in the two Yu Pagodas.  They heal B.F.A. by 1500 each, cancel all status problems on him, and boost his Overdrive Gauge by about 1/6.  Use Wakka to deal with them during the fight with B.F.A.  They move into Melee range when facing Yu Yevon.  Be careful, because if one is moving and the other one isn't, the one moving will inflict Darkness, Silence, Sleep, Poison, and Curse all in one move.  If you use Wakka's Overdrive right, you can usually knock them both out in one blow, but they start moving later.  Also, each time you stop them, they come back with the TOTAL damage that was caused to them last time as their new HP.
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Braska's Final Aeon-Phase 2
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This one is much more difficult than the first one.  Once you've taken him down the first time, Braska's Final Aeon comes back even more powerful than before.  His HP doubles to 120,000 for one thing.  He also has a new weapon: The Eternal Blade (yes, that name is made up).  With this blade, Braska's Final Aeon can attack all three of your party members at once.  Each swing does around 3,000 damage to each character.  Also, his Triumphant Grasp doubles in attack power.  This battle is much more difficult, especially if you've been focusing on the Yu Pagodas.
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Braska's Final Aeon-Final Phase
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When B.F.A. gets below 60,000 HP, he unleashes his most devastating technique: "Ultimate Jecht Shot" which can do 5000-6000 damage to each party member.  To counter this, use Tidus's command "Talk" to bring B.F.A.'s Overdrive Gauge back down to zero.  You can only do that twice in the entire battle, so wait until this phase to use it.  Afterwards, just have Yuna summon one of her Aeons to take the blow for you.  Do NOT fight with Aeons, because the Yu Pagodas will become a HUGE pain if you do.  Just keep using Auron and Tidus to attack B.F.A. and have Wakka attack the Yu Pagodas until B.F.A. is dead.  Now comes the fun part.
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Possessed Aeons
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Once Braska's Final Aeon is defeated, Yu Yevon will come out of Jecht and transport your group to an alternate dimension.  Here you must have Yuna summon every Aeon you've collected throughout the game.  When you summon one, Yu Yevon will possess it and you'll have to defeat it.  Fortunately, you have two advantages.  First, everyone in your party is given a free Auto-Life status, so if you are killed you will come back to life.  Second, the Aeons have the exact same stats as they did when you used them.  Same strength, same attacks, same HP.  I killed each one of them with just one shot from Auron.  This continues until every last Aeon is defeated (which really doesn't take that long).
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Final Showdown: Yu Yevon
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This is it.  The final battle.  Yu Yevon is actually pretty weak.  He has 99,999 HP and has no physical attacks.  His magic, however, is devastating.  Before doing ANYTHING, cast Reflect on Yu Yevon.  Whenever he is attacked, Yu Yevon will automatically cast Curaga on himself, healing 9,999 HP each time.  If Reflect is cast on him, the magic bounces off onto you!  Handy isn't it?  Still, the Yu Pagodas are still there to annoy the heck out of you.  Thankfully, they are now in melee range and easy to stop.  Yu Yevon has many powerful magic spells that he'll use on his official turn.  When his health gets low (about 20,000) he casts Ultima, which could mean instant death for anyone under 10,000 HP.  However, you're still under Auto-Life, so no need to worry.  Just keep hammering away at Yu Yevon until he's dead.  Congratulations, you've beaten the game!