Eidolons

Garnet, as the main summoner of the group, is able to summon Eidolons (known as summons, espers, guardian forces, etc. in previous Final Fantasy installemts) to aid the party in battle. She cannot summon during disc one (well, she can, technically, but they require a ridiculous amount of MP). Afterwards, in disc two, she can summon normally. Garnet's summoning abilities make her special and is why Queen Brahne wants the capture of the young princess, so she can snatch the powerful Eidolons and use them for herself!

The game has more than eight Eidolons, but Garnet has eight Eidolons that can only be used by her. Everytime Garnet gets a new gemstone, it means she can summon a new Eidolon. They include -

Shiva
Element: Ice
Gemstone: Opal
Attack: Diamond Dust

Ifrit
Element: Fire
Gemstone: Topaz
Attack: Flames Of Hell

Ramuh
Element: Lightning
Gemstone: Peridot
Attack: Judgement Bolt

Atomos
Element: Gravity
Gemstone: Amethyst
Attack: G-Force 199

Leviathan
Element: Water
Gemstone: Aquamarine
Attack: Tsunami

Odin
Element: Non-elemental
Gemstone: Dark Matter
Attack: Zantetsuken

Bahamut
Element: Non-elemental
Gemstone: Garnet
Attack: Mega Flare

Ark
Element: Dark
Gemstone: Pumice
Attack: Eternal Darkness

The more gems of one stone that you own, the more powerful that particular eidolon will become. As well, the first time you summon an eidolon, you can watch the entire summon scene. A long summon scene means a powerful attack. However, the chances of the entire summon scene playing after the first try is lower. If only a short summon scene shows up (basically the Eidolon doesn't appear; only its attack is shown), it means it is a weaker attack. There is a special item you can use to make sure that a long summon scene always plays to ensure strong attacks.