Gauge increase : Your Theurgy gauge will gradually increase after summoning a Persona

The Protagonist has 7 Theurgy skills, you unlock these by having a different combination of personas.

• Cadenza (Orpheus + Apsaras): Restores 50 percent HP to all allies while increasing their accuracy and evasion stats.
  
• Jack Brothers (Jack Frost + Pyro Jack): Deals medium Almighty damage to all enemies with a high chance of inflicting Down.
  
• King and I (Black Frost + King Frost): Deals heavy Ice damage to all enemies with a medium chance of inflicting Freeze.
  
• Best Friends (Forneus + Decarabia): Places a buff that boosts an ally’s attack, hit/evasion and defense rate, this way applying the effects of Tarukaja, Rakukaja, and Sukukaja skills.
  
• Scarlet Havoc (Siegfried + Mithras): Deals severe Slash damage to all enemies while ignoring all resistances.
  
• Trickster (Susano-o + Loki): Deals massive Almighty damage to all enemies with a chance to inflict a random status ailment.
  
• Armageddon (Helel + Satan): Deals ultimate Almighty damage to all enemies in exchange for reducing your HP and SP to one.

Every Theurgy skill for The Protagonist has a different affect this means that they can used in a variety of different situations, additionally it is very easy to increase your Theurgy gauge due to the fact that all the protag has to do is summon a persona.
The best Theurgy skill that the protagonist can have is Armageddon, it will take a while to get as you don't get access to Helel and Satan for a while but when you do get them it is a very strong skill to have.

Gauge Increase : Will gradually increase after using a healing skill

Yukari's primary role is to heal the team and use wind attacks

• Cyclone Arrow (Io): Deals severe Wind damage to an enemy while ignoring their resistances.
  
• Tranquility (Isis): Places a buff that ensures the next magic attack used by any party member will deal more than double damage.

Cyclone Arrow will be great for any enemies that have a wind weakness but will still be useful for almost any enemy as it ignore resistances.
Tranquility is more focused on buffing the team rather than doing damage.

Gauge increase: gradually increase after landing a critical hit.

Junpei is mostly focused on physical damage with a little bit of fire damage on the side.

• Hack n’ Blast (Hermes): Deals severe Slash damage to an enemy while ignoring their resistances.
  
• Blaze of Life (Trismegistus): Deals massive Fire damage to an enemy while ignoring their resistances. This skill also fully heals the user.

Hack n' Blast is more focused towards melee/Slash damage while Blaze of Life is more focused towards fire and healing.

Gauge increase : gradually increase after using a buff skill on himself.

Akihiko is kind of like Junpei, he has a focus on physical and a secondary focus on an element, in this case it's electricity.

• Lightning Spike (Polydeuces): Deals heavy Electric damage to all enemies while ignoring their resistances.
  
• Electric Onslaught (Caesar): Deals severe Electric damage to an enemy while ignoring their resistances. This skill also has a high chance of inflicting Shock.

Lighting Spike is focused on elemental damage to all enemies where Electric Onslaught is focused on a single enemy.

Gauge increase : gradually increase after debuffing enemies or inflicting them with a status ailment

Mitsuru is focused on ice attacks and debuffing the enemy

• Blizzard Edge (Penthesilea): Deals severe Ice damage to an enemy while ignoring their resistances.
  
• Blade of Execution (Artemisia): Deals massive Almighty damage to an enemy while decreasing their stats.

Blizzard Edge is a very good ice attack where Blade of Execution is an Almight attack which also decreases the enemies stats.

Gauge increase : gradually increase after analyzing an enemy

Yamagishi does not contribute to combat but is very good support

• Oracle (Lucia): Places a random buff on all party members.
  
• Revelation (Juno): Places a random enhanced effect on all party members.

Yamagishi is all about buffs, normal or enchanced.

p.s you can't say Fuuka on steam without it being censored so i'm saying Yamagishi

Gauge increase : gradually increase after using physical skills.

Aigis is a very good physical attack user

• Orgia Mode (Palladion): Deals heavy Pierce damage to all enemies while ignoring their resistances. Aigis cannot be controlled for the next few turns.
  
• Maximum Firepower (Athena): Deals severe Pierce damage to an enemy while ignoring their resistances.

Orgia Mode will do a lot of damage but not allow Aigis to move for a few turns after that.
Maximum Firepower does a lot of Pierce damage to an enemy.

Gauge increase : gradually increase after exploiting an enemy's weakness.

Koromaru is focused mainly on physical buffs and dark damage.

• Hound of Hades (Cerberus): Deals severe Dark damage to an enemy while ignoring their resistances.
  
• Power Howling (Cerberus): Places a buff that ensures the next physical attack used by any party member will deal more than double damage.

Hound of Hades is a good Dark damage attack, Power Howling is good for helping your party's attack

Gauge increase : gradually increase after his SP drops below half

Ken is mainly a light type damage dealer.

• Divine Retribution (Nemesis): Deals severe Light damage to an enemy while ignoring their resistances.
  
• Divine Intervention (Kala-Nemi): Revives all party members and fully heals them. The next attack used against any teammate will be reflected.

Divine Retribution is a good Light type attack but Divine Intervention is a very good healing and revival skill.

Gauge increase : gradually increase when HP is below half

Shinjiro is also a powerful physical attacker

• Bleeding Fury (Castor): Deals severe Strike damage to an enemy while ignoring their resistances.

Bleeding Fury is a very good Strike damage dealer

Thanks for reading this, hope you don't die in Tartarus now!
Feel free to like or favourite the guide if you found it helpful.

Credits to Dot Esport for all the descriptions :
https://dotesports.com/persona/news/all-theurgy-powers-in-persona-3-reload