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