Basic summons, known as Eidolon's, have returned to Final Fantasy IX. Both Eiko and Dagger are the only characters that can call upon these beasts for aid.
To learn a Eidolon, you need to obtain the right Jewel for it, then get the AP needed. While you have the Jewel equipped, you can summon the Eidolon, without having gained the total amount of AP. But if you un-equip the Jewel before learning it, you will no longer be able to call upon that Eidolon.
Also, the power of the Eidolon depends on the summoning scene. The full version causes more damage, whereas the short version does less damage. To get the full summon each time, you will need to learn the "Boost" ability. This is learned from a Pumice Piece at 190 AP with Dagger, and 150 AP with Eiko.

Dagger's Eidolons

	Shiva
	AP	20	MP	24
	Attack	Diamond Dust
	Element	Ice
	Description	Attack against all enemies
	Power	36 + number of Opals in the inventory
	Jewel	Opal

	Ramuh
	AP	30	MP	22
	Attack	Judgment Bolt
	Element	Lightning
	Description	Attack against all enemies
	Power	42 + number of Peridots in the inventory
	Jewel	Peridot

	Ifrit
	AP	35	MP	26
	Attack	Hellfire
	Element	Fire
	Description	Attack against all enemies
	Power	42 + number of Topazes in the inventory
	Jewel	Topaz

	Atomos
	AP	30	MP	32
	Attack	G-Force 199
	Element	Lightning
	Description	Proportional damage to all enemies.
	Power	Target's max HP x (number of Amethyst) + 30
	Jewel	Amethyst

	Odin
	AP	30	MP	28
	Attack	Zantetzuken
	Element	-
	Description	Instant KO (Or damage with Odin's Sword)
	Power	145 + number of Ores in the inventory
	Jewel	Dark Matter

	Leviathan
	AP	40	MP	42
	Attack	Tsunami
	Element	Water
	Description	Attack against all enemies
	Power	59 + number of Aquamarines in the inventory
	Jewel	Aquamarine

	Bahamut
	AP	80	MP	56
	Attack	Mega Flare
	Element	-
	Description	Attack against all enemies
	Power	59 + number of Garnets in the inventory
	Jewel	Garnet

	Ark
	AP	100	MP	80
	Attack	Eternal Darkness
	Element	Shadow
	Description	Attack against all enemies
	Power	-
	Jewel	Pumace

Eiko's Eidolons

	Carbuncle
	AP	35	MP	24
	Attack	Ruby Light
	Element	-
	Description	Casts Protect & Reflect
	Power	-
	Jewel	Ruby

There are 4 different actions for Carbuncle. The action depends on the Jewel equiped at the time.
Jewel	Name	Spell Cast
No Jewel	Ruby Light	Reflect & Protect
Emerald	Emerald Light	Haste & Protect
Moonstone	Pearl Light	Shell & Protect
Diamond	Diamond Light	Protect & Vanish

	Fenrir
	AP	55	MP	30
	Attack	Terrestrial Rage
	Element	Earth
	Description	Attack against all enemies
	Power	42 + number of Sapphires in the inventory
	Jewel	Sapphire

Fenrir also has two different Attacks, depending on what item you have equiped.
Item	Name	Damage
No item	Terrestrial Rage	Earth Attack
Maiden Prayer	Millennial Decay	Wind Attack

	Phoenix
	AP	40	MP	32
	Attack	Rebirth Flame
	Element	Fire
	Description	attacka all enemies while reviving KO'ed characters with some HP.
	Power	40; recovers [target's max HP x (target's str + 30)]% HP
	Jewel	Phoenix Pinion

	madeen
	AP	120	MP	54
	Attack	Terra Homing
	Element	Holy
	Description	Damage to all enemies
	Power	71+ Eiko's level
	Jewel	Ribbon

Dagger and Eiko's Eidolon

There is one other Eidolon in Final Fantasy IX worth mentioning. This Eidolon is known as Alexander. He cannot be summoned by Eiko or Dagger individually, but rather is summoned by both together. It only happens once in the game, and is completely story-related. It is also the last time we will see this awesome summon to date.