To acquire these summons, you have to get the Summoner asterisk first. Refer to this link to learn how. Take note that the summon you acquire can also be used by the Conjurer job class.
To get the summon, you only have to survive the summon's attack for a turn. Take note that this attack is very powerful and can wipe out your party instantly without proper preparation. To get the summon, you only need at least one person to survive the attack.
This is the default summon that you can cast after getting the Summoner asterisk.
Location: Temple of Wind
Talk to the anchorite of wind, mouthpiece of Hresvelgr in the temple's second floor. You have to withstand the attack from that summon before you gain command of it. You need at least one party member to survive the attack, in order to complete the challenge. To do this, equip the abilities that will increase magic defense and have that particular character carry a Wind Charm. The higher the magic defense, the better.
Location: Mt. Fragmentum - Slope
This person should be at least one of the magician classes, preferably a white mage that has access to the following support skills: Abate Water and M. Defense 10% Up
If you have the Mage Shell or Ice Charm, equip it to that character as well. The idea is to concentrate all possible magic defense boosts and water-type damage reduction abilities to that character. And that character should have at least 1000 HP. These preparations should be enough to survive Ziusudra's Sin.
Location: Temple of Fire
You can find the anchorite of fire, mouthpiece of Promethean Fire in a small room to the southeast. You should know the drill already; equip your character with accessories and abilities that can resist fire damage like the Flame Charm you acquired earlier or the Black Mage support ability Abate Fire. There's also an Iceflame Shield that you can get from the chest beside the anchorite. This nifty equipment will make you immune against fire and water damage. Make sure to save your game first before attempting the challenge.
Location: Everlast Tower, 2F
In the second floor, head to the east to find the anchorite of bolts, mouthpiece of Deus Ex. This is a lightning-based summon so equip one character with Thunder Charm or Abate Lightning (Summoner JLV 2) The latter is preferred since you can use the accessory slot for something else, like Force Armlets, Mage Shell or Barrier Shroud. Get his/her magic defense as high as you can and make sure you have high HP. Don't bother using the Reflect Ring since summon magic can't be blocked by Reflect.
Crescent moon-shaped island to the northeast of the world map. (Available only in Chapter 5)
Fly to the crescent moon-shaped island to the northeast of the world map and land on the ground. Walk to the west and follow the land to the northwest until you trigger a scene and find the anchorite of war.
Angelic Ward will play a great role here, as well as Absorb Magical Damage. He can deal tremendous non-elemental magic damage and can chip off a huge chunk of your HP. If your party gets wiped out after several attempts and adjustments, come back later after earning a few more levels and getting more equipment. You need to raise your magical defense as high as you can, through accessories and support abilities.