In Summon Night 5, There are various classes that can be unlocked by either leveling up or certain requirements in the game like completing a mission or upgrade your weapon's level.

High Summoner (高等召喚師) - Balanced Unlocks at level 12
Armed Mediating Officer (武装調停官) - Physical offense Complete Mission No.11 (unlock by having 50 physical kills with Folth and Arca)
Special Resonance Officer (響命特務官) - Magical offense Complete Mission No.12,(Unlock by having 50 magic kills)
Protector of Bonds (絆の護り手) - Defense Unlocked by completing mission 33 which is unlocked by completing 25 missions
User of Forbidden and Disastrous Arts (禁厄術士) - Balanced; karma class Unlocked by having 3 or more diamonds of karma before the first battle in Chapter 13. Talk to your Cross at the Park of Meetings.

Machine Soldier of Steel (鋼鉄機兵) - Physical offense   Unlocks on Lv 14
Machine Soldier of Flashing Magic (閃の魔導機兵) - Magical offense Unlocked by killing 30 enemies with skill ability(ranged spell skill) varies by partner.
The Gentle Superman of Lightning and Steel (優しき電鋼超人) - Defense 50 bonus points or more spent in DEF and MDF.

Boy of Oni Flames (鬼焔童子) - Balanced  Unlocks on Lv 14
Master Swordsman of Offensive Fire (撃炎剣豪) - Physical offense 30 bonus points or more spent in TEC.
Crimson Priest (紅蓮法師) - Magical offense 30 kills with Kagerou's ranged spell.

Guardian Angel (守護天使) - Magical offense  Unlocks on Lv 14
The Eternal Shining Angel (悠久たる閃天使) - Physical offense Upgrade Spinel's weapon's ATK 5 times.
The Angel of the Stars Who is Called Infinite (無限なる星天使) - Magical offense 30 kills with Spinel's ranged spell.

Starving Princess (はらぺこプリンセス) - Physical offense  Unlocks on Lv 14
Magical Queen (まじかるクイーン) - Magical offense 30 kills with Pariet's ranged spell.
Beatdown Queen (ぽかぽかクイーン) - Defense 50 bonus points or more in DEF and MDF.

Special Magic Police-Knight (魔導特警士) - Magic offense Unlocks at Lv 13
Flashing Swordsman on a Mission (特命閃剣士) - Physical offense Beat Mission 21 (Unlocked by reaching Chapter 8)
Police-Knight Detective of the Gale (疾風の隠密警士) - ???

Beat Mission 36

Special Duty Knight of the Sakura Flowers (櫻花の特務騎士) - Magic offense Lv 13
Special Duty Knight of the Silver Stabs (銀穿の特務騎士) - Physical offense Clear Mission 21 (Unlocked by reaching Chapter 8)
Knight Who Protects the Dragon God (龍神守護騎士) - Defense Clear Mission 34


High Summoner (高等召喚師) - Balanced Lv 13
Onslaught Lady (猛攻淑女) - Melee Upgrade Cyda's weapon proficiency to LV3
Guardian of the Secret Formula (秘法の守護者) - Magical offense Beat Mission 35 which can be unlocked by beating Mission 34
High Summoner (高等召喚師) - Balanced Lv 14
Aggressive Special Officer of Iron (鉄攻特武官) - Physical offense Upgrade Calis's weapon's ATK five times.

Tricky Officer on a Mission (機巧特命官) - Magical offense

Kill 50 enemys with spells.

High Summoner (高等召喚師) - Balanced Lv 13
Technique User in a Blood Pledge with the Great Youkai (大妖の血盟術師) - Physical offense Perform 50 debuff spells on enemys
Armed Monk of the Ferocious Youkai (暴妖の荒法師) - Magical offense Kill 50 enemys with spells

Class 1 - Balanced  Lv14
Class 2 - Offense Defeat 50 enemys with physical attack.
Class 3 - Magic 50 bonus points or more spent on MAT and MDF

Class 1 - Magic Lv15
Class 2 - Range Upgrade Torque's gun proficiency skill to LV3
Class 3 - Defense 50 bonus points or more spent on MDF and DF
Class 1 - Magic LV27


Class 1 - Magic LV27