Nix is a tool and a functional language that takes a unique approach to package management and system configuration.
Nix also employs a clever way to store packages/derivations/data to reuse and avoid building software on demand (instead pulling from the nixpkgs registry if possible).
NixOS
NixOS is an extension of Nix, basically applying nix as the replacement for traditional tools like apk, apt, pacman, etc…
What nixos does above these tools is that it also allows granular configuration of the OS itself. This means you add packages at the same place you add/edit configs, making it super easy to reproduce your environment.
I think of nixos as ansible on steroids, that you install on the host machine rather than a remote (with which you SSH into like ansible does). You also get the benefits of the nix caching/store described above.