How to get the Leviathan Summon in FF4
Leviathan is one of the two optional Summons that you can acquire in the Land of Summons. For more information about how to reach this location I would recommend you read my FF4 Chapter 9 Walkthrough page. In order to challenge Leviathan to a battle you'll first need to defeat Asura, the boss standing next to Leviathan. Fore more information about Asura, head over to my FF4 Asura Summon Walkthrough.
Unlike with Asura, the fight against Leviathan will be very easy. He has a lot of HP but he works like an average boss with no unique strategy. Just make sure that after you beat Asura you head upstairs and use a Cottage before challenging Leviathan.
The signature move Leviathan has is called Tidal Wave and it will hit your entire group for about 700 - 800 damage. He will also occasionally use Blizzara which only hits for about 400. Leviathan has a slower than average attack speed which will give you plenty of time to heal up between attacks. Also, he is highly susceptible to Lightning magic so blow him up with Rydia's most powerful spells.
Every round of combat you'll want to use Curaga on your entire group with Rosa and Jump on Kain any time he is able to. Jumping with Kain will allow him to avoid all of the AoE damage that Leviathan does and still do damage himself.
Once Leviathan is defeated Rydia will learn him as a summon and you're free to leave the Land of Summons. I'd recommend you return to Chapter 9 of my FF4 Walkthrough to finish collecting the remaining treasures in the Land of Summons and information on how to leave.