Gen. 1 chains boost damage of the corresponding element by 20% and offer a chain bonus up to 99 hits. They typically deal 11 hits and provide a minor party buff, although some exceptions exist. They typically come in both physical and magical versions, although some elements never had a magical variation released.
Gen. 2 chains boost damage of the corresponding element by 50% and offer a chain bonus up to 150 hits. They come in both physical and magical versions; the physical version boosts ATK by 50%, while the magical version boosts MAG by 50%. (The holy magical chain is the exception; since some holy damage is MND-based, it gives a MAG/MND boost of 30%.) These damage and stat boosts make gen 2 chains a significant boost over gen 1.
Gen. 0.5 chains boost damage of the corresponding element by 20% and offer a chain bonus up to 99 hits. They have no other effects. They were made freely available to all players in a series of events some time after gen. 1 and 2. They're typically available on hybrid characters (to cover both physical and magical).
Gen. 2.5 chains boost damage of the corresponding element by 50% and offer a chain bonus up to 99 hits. They're instant cast, boost party members' elemental damage (on top of the 50% field effect), and have the same stat boosts as gen 2 chains. They ramp up faster than gen 2 chains but have a lower damage cap, so they're not universally better (hence the "2.5" moniker), but having less drop-off when renewing a chain and earlier savage mode breaking is often beneficial.
A RW (Roaming Warrior) chain, Fabula Sorcerer, is available in magicite and Odin battles. Like the gen. 0.5 chains, it boosts damage of the corresponding element by 20% and offer a chain bonus up to 99 hits but has no other effects.
Element
Gen. 0.5
Gen. 1
Gen. 2
Gen. 2.5
Phys.
Mag.
Phys.
Mag.
Phys.
Mag.
Fire
Elena
Locke
Krile
Gilgamesh
Auron
Rubicante
Vincent
Balthier
Vivi
Ice
Fran
Snow
Rinoa
Laguna
Celes
Serah
Edea
Fran
Rinoa
Lightning
Reno
Kain
Shantotto
Prompto
Shelke
Desch
Garnet
Queen
Shantotto
Earth
Rude
Ingus
Machina
Galuf
Red XIII
Emperor
Gladiolus
Tellah
Wind
Freya
Zack
Faris
Luneth
Alphinaud
Fujin
Vaan
Barbariccia
Water
Edge
Tidus
Paine
Rikku
Gogo (V)
Strago
Yuffie
Meia
Holy
Cecil (Paladin)
Warrior of Light
Ceodore
Ramza
Edward
Hope
Basch
Yuna
Dark
Vayne
Garland
Seymour
Seifer
Gabranth
Cloud of Darkness
Golbez
Shadow
Kuja
Poison
Quistis
Realm Chains
Gen. 1 chains boost damage of the corresponding realm by 50% and offer a chain bonus up to 150 hits, along with providing party-wide buffs of Haste and +30% ATK and MAG.
Gen. 2 chains boost damage of the corresponding realm by 50% and offer a chain bonus up to 99 hits. They're instant cast and provide party-wide buffs of +30 ATK/DEF/MAG/RES and instacast 1. They were intended to counter the Full Break effects that higher-end Cardia bosses use, but they're of limited use for that; restarting a chain can hurt your damage almost as much as a Dreambreaker Full Break. They remain useful as general party buffs.
A RW (Roaming Warrior) chain, Bonds of Historia, is available in Cardia battles. It boosts damage of the corresponding realm by 30% and offer a chain bonus up to 150 hits but has no other effects.