Classes in General

Unless specified for an individual campaign, all classes are allowed.  The official Pathfinder Classes can be found here.
People are not aware of "classes" for the most part, but the specialized or well educated do understand the differences in capabilities that different training allows, and some classes are formalized (Alchemists, Druids, Wizards, Bards, Clerics and Champions) by organization, affiliation or specific methodology. For these classes, those in the know will be able to tell what you are if they see you using your abilities. For others, people may generalize based on your equipment or capabilities.
Fighter, Swashbuckler, Gunslinger, Monk: People will see you as very capable and may turn to you for help or do their best to stay out of your way, depending on their first impression of you. Unless you are affiliated with a military, people will likely assume that you make your living my hunting monsters or bounties.
Ranger: As above, but you will be seen as a master of the wild as well.
Barbarian: As Fighter, but people are more likely to react to you with fear rather than admiration.
Champion: As Fighter, but you will likely also be seen as someone who can administer aid and act on behalf of your church in a pinch.
Alchemists: People will view you as almost an artisan turned adventurer, and you will be welcome in most towns as a supplier of healing and tools. Some might wonder why you don't settle down and make a shop.
Bard: People will just assume you are a travelling performer unless you show them otherwise. If you are especially talented they may make the connection that you are a formally trained Bard.
Cleric: People will see you as a representative of the church and someone who can act on it's behalf.
Druid: People will see you as a master of the wilderness, and someone who can assist with their problems related to nature, though they may be wary of you if they are more in tune with the civilized world.
Sorcerer, Witch, Oracle: People assume you are a wizard, cleric or other magical class.
Wizard: People will see you as a bit strange, but powerful. And will likely assume that you are standoffish or arrogant.
Magus, Summoner, Inventor: Rare enough that people will not have a predetermined reaction.