Vesper's goal is to curate ideas, to give thoughts and other things a place where they can grow and become what you want. To organize your notes, Vesper uses tagging and archiving along with a drag-and-drop hierarchy.
I think the best way to describe the archive is as Q Branch did in the app's description: "When you’re done with an item, swipe it to send it into the archive. Out of sight, but remembered forever".
Tagging is the core of Vesper, to curate your notes, you can tag them to create lists that are like iTunes Smart Playlists, you can have a note in a few different tags. You can curate your notes however you like, but I really like the subtle suggestion the app makes to point you in the right direction, like when you have too many tags for a note or tags with really long names, the app won't show all of them.
How you use Vesper is up to you, and I use it to capture all of my ideas and as a place for all the things don't have another place they belong to (yet), like a phone number I need just for the day, or a to-do list of tasks for today like take out the trash, that don't belong in Clear or OmniFocus.
Rene Ritchie of iMore, concludes his review of Vesper saying: "It's a Cocktail, not a Soda", and I think it is a great summary of Vesper (named after James Bond's favorite cocktail, with the recipe in the credits). Vesper is opinionated, and without sync and other features most people are looking for a note taking app, it might not be for everyone.
Vesper is available for 4.99$ on the App Store.