Here is the full list of all 73 Planar Conquest achievements.
Explore a world feature
Cast a spell
Recruit a unit in one of your cities
Forge an artifact
Research a spell
Raze a city
Kill an enemy hero on the world map
Recruit a hero
Banish an enemy SL
Win as Eyegor
Make an alliance with another SL
Have a unit deal fire, cold, electrical and acid damage with one attack
Kill a Drake with a Settler unit
Capture a city
Win as custom SL
Win by casting Spell of Domination
Achieve tier 9 in all elemental circles
Win the game by defeating all SLs
Found a new city
Win as R'jak
Win as Grey Elves
Research all available spells
Win as High Men
Win as Unhallowed
Bless a Water Elemental
Return from banishment
Win as Rastiss
Create a Custom SL
Win as Ariel
Win as Dark Elves
Make a trade with another SL
Construct all buildings in a city
Have six max level heroes with all of their equipment slots filled
Have a city of three living races
Win as Alhasret
Win as Neve
Enchant a unit with five unit enchantments
Travel to another plane
Have at least 100k population in your empire
Win as Istimaeth
Summon a champion
Win the game without casting any spells
Defeat a SL
Win as Markas
Reach 100 spellcraft
Have a hero reach maximum level
Win a game having spent (or gained) no points in SL creation
Win as Merlin
Win a battle
Reach 100 fame
Have a unit reach maximum level
Take control over every city on Air plane
Conquer Hell Gateway feature
Visit earth plane by going through Earth Portal
Build a city on water plane
Conquer a city with army commanded by Vendral
Win as Draconians
Win as Drakhis
Win as Orcs
Win as Magog
Win an encounter with army commanded by Bonebasher
Defend your city with army commanded by Kjorgi
Win as Dwarves
Win as Varkin
Cast a spell with Draylik
Win as Myrodants
Win as Prithvi Mata
Add Elven Hussar to your army
Cast a spell with Ogre Mage
Reach maximum level with Nehanash
Win as Erzsebet
Recruit The Last Uruthian
Recruit Aelys the Broken