Old school vancian? Uggh.
Oh well, I have 1e friends who will be happy.
At least there are other nonvancian classes to choose from?
PF2 is similar to 3e in this regard. Clerics, druids, and wizards are old-school Vancian (albeit with unlimited cantrips), and bards and sorcerers are spontaneous casters with a small repertoire that they can cast in a more flexible fashion.
But are these nonvancian classes as appealing as the Wizard is, in terms of versatility, effective spells, and game support − or is the Wizard "privileged" like it is in 3e?
The core rules have two spontaneous casters. The bard uses the Occult spell list, which is fairly versatile but light on physical damage. The sorcerer can use any of the four spell lists, depending on their bloodline – draconic sorcerers use arcane magic, fey or elemental ones use primal magic, angelic or demonic ones divine, and aberrant ones occult. In addition, they gain one spell per spell level "for free", sometimes poaching them from other spell lists (e.g. fey sorcerers get some mind-affecting stuff, which is otherwise not primal magic's forte).
Spontaneous casters are more flexible at the cost of long-term versatility. They are strong offensively (because they can cast the same spell repeatedly), but won't have many niche spells. That said, my primal sorcerer has gotten a lot of mileage out of some utility spells like
speak with animals and
wall of stone.
What is an "archetype"? Is it something like a multiclassed class or a prestige class?
An archetype is akin to a multi- or prestige class.
The way they work mechanically is that each archetype has a Dedication feat, e.g. Pirate Dedication, Magaambiyan Attendant Dedication, or Wizard dedication. This is always a class feat, so if your wizard wants to be a Magaambiyan Attendant they take the Magaambiyan Attendant Dedication feat instead of a Wizard feat. They still gain the regular class progression (proficiencies, spellcasting, etc.), but instead of learning a special wizard trick, they learn the dedication feat.
Each archetype also has a number of other feats that delve deeper into the archetype. These are often, but not always, also class feats (some are skill feats). Generally, you can't take a second dedication feat unless you've taken at least three feats in your first archetype, but some archetypes build on one another and bypass that rule (e.g. Halcyon Speaker builds on Magaambiyan Attendant, so you don't need to "finish" the Attendant archetype before taking the Speaker dedication).
The core book only has the multi-class archetypes. The Advanced Player's Guide has a lot of "general" archetypes (e.g. Pirate, Medic, Acrobat), while more flavorful and setting-based ones show up in other books – particularly in the Lost Omens line.