Using their own Next.js as an example: everything in /pages/ is client app, everything in /api/ is serverless app.
Client talks to the /api/ endpoints, /api/ endpoints talk to your domain services.
Vercel also provide edge-location hosting and caching.
They have a suite of complementary software products and services, including NextJS (a "batteries included" React framework with robust support for SSR and SSG), Now (hosting / CDN / runtime in "serverless" paradigm), Micro, etc.
World-class DX and remarkably powerful tools that somehow combine simplicity and flexibility.
No affiliation, just a longtime grateful fan.
FWIW I liked "Zeit" better than "Vercel" (shrug). But preferences aside, I was annoyed that the new redirects for "now.sh" URLs were -- uncharacteristically for a team famous for attention to detail -- mishandled, e.g. breaking a fantastic preso I'd bookmarked (which grauch used in his React Conf 2017 talks) and which I had cause to reference today in promoting NextJS with a client. It's the only real "footgun" move I've seen them make. Here's hoping it's an anomaly!
"Serveress" = you aren't managing a logical server
I.e. you don't need to concern yourself with provisioning, nor maintaining, a server as that part of the process has been abstracted away from you.