Notarization is a security check that Apple runs on macOS software that developers distribute outside the Mac App Store. The Apple notary service is automated; it is not App Review. It works together with Gatekeeper, the part of macOS that checks downloaded software before it first runs, and with Developer ID, the code-signing certificate that identifies the developer.
The developer signs the app with a Developer ID certificate, turns on the Hardened Runtime and uploads the app, disk image (DMG), installer package (PKG) or ZIP file with Xcode or the notarytool command-line tool. The service scans it for malicious content and code-signing problems. If there are no issues, it issues a ticket, a record that Apple checked the software; the developer can staple the ticket to the file with stapler, and Apple also publishes it online. On first launch, Gatekeeper finds the ticket and tells the user that Apple checked the software. Since macOS 10.15, all software built after June 1, 2019 and distributed with Developer ID must be notarized.
There is no separate fee; a Developer ID certificate requires membership in the paid Apple Developer Program.