Classes are broadly defined specializations of a character. Pathfinder: Kingmaker currently includes all the core classes, plus the Alchemist, Inquisitor, Magus and Slayer. The Kineticist class was added in the The Wildcards DLC.
Archetypes are variants of classes meant to provide more options for character building and development. Each class has three available archetypes.
Prestige Classes are a sort of advanced class that each have their own prerequisites.