Shiva-Found at Squalls study panel in his classroom. Its in the "Tutorial" section.

Ifrit-Found at the end of Fire Cavern.You must defeat the GF for it to join you.

Siren-You must draw Siren during the boss fight with Elvoret on top of Dollet Communication Tower.

Brothers-You must go to the Tomb of the Unknown King.Inside you must complete some tasks and defeat the brothers and they will join you.

Diablos-Get the cursed lamp from Cid. Use the lamp and you will go in battle with it defeat it to get him to join you.

Carbuncle-You must draw it during the fight with the Iguions(1st disc)

Leviathan-You must draw it from the boss NORG at balmb garden(2nd disc)

Pandemona-You must draw it from Fujin during the boss fight with her and Raijin(2nd disc)

Cerberus-You can challenge it in the main hall of Galbadia Garden (2nd disc) You have to defeat it for it to join you (this is an optional battle)

Alexander-You must draw it from the Edea fight in Galbadia Garden (2nd disc)

Doomtrain-You first have to get the Solomon Ring at Tears Point. Once you get it you have to collect these items to summon it here are the items:six Malboro Tentacles,six Remedy+,and six Steel Pipes.Select the ring in your items and it will join you with out a fight.

Bahamut-When you have the Ragnarok on the third disc fly to the Deep Sea Research Center in the southwest coner of the world map. Inside go to the core in the middle of the room and answer its questions correctly and defeat it and it will join you.

Jumbo Cactuar-Go to Cactuar Island a desert island east of the Centra Ruins. You'll see a green figure poping in and out the sand go to the green figure fight the big cactuar defeat it and it will join you.

Tonberry King-You must fight 18~22 tonberrys at the Centra Ruins. After you defeat the tonberrys the tonberry king will come out defeat it and it will join you.

Eden-After defeating Bahamut you'll be able to go deeper into the Research Center complete the tasks that need to be done. You'll fight a thing called Ultima Weapon make sure you draw eden from it.