Neon is a managed serverless Postgres offering.

"Managed" means we take care of Postgres administration.

"Serverless" means that your database isn't running if you're not using it. We put computes to sleep after a certain amount of time. We can also scale database resources up and down as needed.

(Neon engineer)

I think it usually means that it's fully queryable by serverless functions, usually via atomic HTTP API calls, and often without network restrictions.

But in this case it seems to be referring to the database itself, and the ability to scale to and from zero resources on demand.