Chain Attacks are basically combo skills that allow you to attack continuously provided you have the required AP to use them and you time each attack correctly.

Chain Attacks are very useful to Summoner's combat system.
Using them will increase your overall damage output and some even give you access to specific damage types that can be far more effective against certain enemy types.
Some Chain Attacks will also provide buffs and healing to a character or even the whole party and some will inflict status effects on enemies.
Using Chain Attacks also locks the targeted enemy down so they can't attack you while the chain is successfully ongoing.
This can even be used to completely deny some enemies the ability to attack you during a whole fight.
They can take a little getting used to but they are a lot easier than they first appear.
So I strongly recommended you learn how to use these very useful abilities while you play through this fantastic RPG.

To use a Chain Attack all you need to to is right click when you see the small chain icon appear above your selected character while in combat.. (it will look like a sideways 8)

Summoner also gives you the option to manually select specific chains using CAPS, SHIFT, CTRL and ALT.
You can change your chain keybinds and select new chains when you unlock them in the Character menu.. they will be under your skills as you will notice when leveling up your characters.

To use a specific chain you need only press and hold CAPS, SHIFT, CTRL or ALT and then right click as usual when the chain icon appears.. it's that simple.
Manually selecting chains is also the fastest way to unlock specific chains as well.

It's important to note that only 4 Chain Attacks can be equipped at any time and those chains under CAPS, SHIFT, CTRL and ALT will be your currently equipped Chain Attacks.

It is also important to note that if you manually only equip 2 or 3 Chain Attacks instead of 4 then you will have to manually select those chains in combat as simply right clicking will not work when less than 4 chains are equipped.

Now that that's out of the way.. lets talk about unlocking new Chain Attacks..

As you regularly use Chain Attacks you'll notice popups saying so and so has unlocked Chain Attack etc etc.

These new chains can be equipped into your Chain Attack rotation the same way you manually change the keybinds in the level up menu.

When you start the game you are automatically given 4 Chain Attacks per character and with exception to Added Blow all other Chain Attacks unlock 2 new Chain Attacks each.
These Chain Attacks are separated by level, 1, 2 and 3 and the speed at which these new chains are unlocked depends on two things.. the specific level of the chain you are using and/or using both of the Chain Attacks in that same line.

Example: Desperation a level 1 chain unlocks Stamina Burst a level 2 chain, but both Desperation and Stamana Burst will unlock Silence a level 3 chain.

You can unlock Silence by only using Desperation or Stamina Burst.. but since Stamina Burst is a level 2 chain it will unlock Silence faster than just using Desperation.

However using both Desperation and Stamina Burst together is the fastest way of unlocking Silence, more so if you use CAPS, SHIFT, CTRL or ALT to use specifically use Desperation and Stamina Burst in in a constant combo spam.. this is by far the fastest method of unlocking a specific level 3 chain.

Right, with all that said and done I'll now go through each character and their specific Chain Attacks.
(Reminder, Added Blow has no second or third chain to unlock)

Level 1 Chain Attacks
Added Blow: Nothing but a standard second attack
Desperation: Deals more damage the less HP Joseph has
Confusion: Attacks the enemy AP (useful against caster type enemies)
Push: Increases duration chance for the next Chain Attack

Level 2 Chain Attacks
Stamina Burst: (Desperation) Deals more damage the more AP Joseph has
Revitalize: (Confusion) Heals the party by a small amount of HP
Mind Drain: (Push) Attacks and Steals a small amount of the enemy AP

Level 3 Chain Attacks
Silence: (Desperation & Stamina Burst) Casts Silence on a enemy (useful against casters)
Burning Hands: (Revitalize & Confusion) Deals fire damage to enemies
Life Leech: (Push & Mind Drain) Steals a small amount of HP from enemies

Level 1 Chain Attacks
Added Blow: Nothing but a standard second attack
Desperation: Deals more damage the less HP Fleece has
Kick: Deals Blunt damage and knocks back foes (highly effective against golems and undead)
Leg Swipe: Chance to inflict Slow condition

Level 2 Chain Attacks
Stamina Burst: (Desperation) Deals more damage the more AP Fleece has
Mug: (Kick) Deals damage and steals a small amount of gold
Arm Slice: (Leg Swipe) Chance to inflict Cripple condition

Level 3 Chain Attacks
Silence: (Desperation & Stamina Burst) Casts Silence on a enemy (useful against casters)
Trip: (Kick & Mug) Trips a enemy causing knockdown increasing all damage taken
Blindness Stab: (Leg Swipe & Arm Slice) Chance to inflict Blindness condition

Level 1 Chain Attacks
Added Blow: Nothing but a standard second attack
Shocking Grasp: Deals energy damage to enemies
Bless: Casts Bless on Rosalind (reduces damage taken)
Confusion: Attacks the enemy AP (useful against caster type enemies)

Level 2 Chain Attacks
Burning Hands: (Shocking Grasp) Deals fire damage to enemies
Revitalize: (Bless) Heals the party by a small amount of HP
Stamina Burst: (Confusion) Deals more damage the more AP Rosalind has

Level 3 Chain Attacks
Icy Touch: (Shocking Grasp & Burning Hands) Deals Ice damage to enemies
Faith: (Bless & Revitalize) Casts Faith on Rosalind (improves magic resistance)
Mind Drain: (Confusion & Stamina Burst) Attacks and Steals a small amount of the enemy AP

Level 1 Chain Attacks
Added Blow: Nothing but a standard second attack
Kick: Deals Blunt damage and knocks back foes (highly effective against golems and undead)
Desperation: Deals more damage the less HP Jekhar has
Mend: Heals Jekhar by a small amount

Level 2 Chain Attacks
Axe Spin: (Kick) Damages nearby enemies with AoE spin attack.. Requires an Axe to use
Sacrifice: (Desperation) Sacrifices some HP but deals double damage
Steel Skin: (Mend) Casts Bless on Jekhar (reduces damage taken)

Level 3 Chain Attacks
Earth Shaker: (Kick & Axe Spin) AoE knocks down nearby enemies.. Requires a Blunt weapon
Life Leech: (Desperation & Sacrifice) Steals a small amount of HP from enemies
Overpower: (Mend & Steel Skin) Casts Empower on Jekhar (Increases damage dealt)

I hope you found this guide helpful :)