Here is the full list of all 14 The Monster Breeder achievements.
Win a fight in the arena
Win a lethal fight in the arena
Insectoid monster reaches level 10
Reptile monster reaches level 10
Carnivora monster reaches level 10
Crossbreed related species
Crossbreed different species
Forge an armor
Forge a weapon
Brew a potion
Build the Monster Lab
Build the Monster Keep
Reputation exceed 50
Reputation reaches 100