Core classes are player classes from the original Pathfinder 1st edition Core Rulebook. Each class was adapted from the original player classes in Dungeons & Dragons 3.5e, and has 20 levels of progression.
Base classes are player classes introduced in later books, such as Advanced Players Guide and Ultimate Magic. They are unique to Pathfinder, and tend to be more complex than the original core classes, but still have similar 20-level progression. Base classes also introduce new thematic elements such as alchemy, firearms, hexes, and undercover intrigue.
Unchained classes are revised versions of the barbarian, monk, rogue, and summoner, introduced in Pathfinder Unchained as an attempt to rebalance issues that were perceived with the original versions of those classes. Having levels in an unchained class counts as having levels in the original version. The unchained barbarian, rogue, and summoner can typically use archetypes intended for their original versions, whereas some of the core monk archetypes don't apply to the unchained monk.
Hybrid classes are player classes introduced in the Advanced Class Guide. Each hybrid class is designed to combine the features of two parent classes, plus their own unique mechanics, and so they tend to be complicated. Having levels in a hybrid class doesn't count as having levels in the parent class, however, some of the hybrid classes may qualify for things that are otherwise available to their parent class. A creature can multiclass with levels in a hybrid class and one or both of its parent classes, although this may restrict certain class feature choices (which are otherwise independent without multiclassing).
While a character can multiclass with these parent classes, this usually results in redundant abilities. Such abilities don’t stack unless specified.
Alternate classes are player classes that mostly consist of the features of one existing class (paladin/antipaladin, rogue/ninja, cavalier/samurai), with a small number of different features. These appear in the Advanced Players Guide and Ultimate Combat books. Having levels in an alternate class does not count as having levels in the original class. Unlike hybrid classes, an alternate class cannot be multiclassed with its original version.
An alternate class operates exactly as a base class, save that a character who takes a level in an alternate class can never take a level in its associated class
Occult classes were introduced in Occult Adventures. Aside from the Kineticist, these options use psychic spellcasting. Psychic spells require mental components (unlike the verbal and somatic components of arcane/divine casting), and some spells can be "undercast" to lower-level versions. Occult spellcasters also use special occult rules, such as occult skill unlocks which allow them to use skills to produce psychic effects.
Prestige classes are player classes with prerequisites. Typically a creature needs multiple levels in one or more non-prestige classes before they qualify for a certain prestige class. Some prestige classes were included in the original Core Rulebook, and others were added over time. Most prestige classes are extremely specialized, and tend to have 10 or fewer levels of progression. Also, all prestige classes use a different progression for base saving throw bonuses.
Class Archetypes are variant packages of class features, which trade some of a class's default features in exchange for more specialized features. Unlike alternate classes, using a class archetype still counts as having levels in the class. Some archetypes borrow class features from another class; for example, the Holy Gun Paladin counts as a Paladin plus some Gunslinger features, but does not count as a Gunslinger. A creature can have multiple archetypes within the same class, unless those archetypes modify or replace the same feature.
NPC classes are intended for humanoid NPCs such as townsfolk, guards, nobles, or even low-level enemies. Creatures with NPC class levels are typically given lower ability scores. NPC creatures may have levels in player classes, but player characters typically don't have levels in NPC classes.