Sibling comment does a good job of going into flakes, but to answer this
> Do you install a package or a service
A package is like the raw software installation. So eg the bash package is just a wrapper around building bash from source (theoretically from source at least... you can also define a package as being a binary distribution as long as you specify the content hash). The service (actually the _module_) is for everything else around software installation. So, eg, the bash _package_ will build bash and have it available for you to put in your path but the bash _module_ might also configure your .bashrc and set it up as your users shell. It would also generally refer to the bash _package_ so you can do all that plumbing but specify a certain version of bash you want to use.
Another common example: a plex package would again build the plex software but a plex module would perhaps create a plex user, setup systemd units, open your firewall, and create a media directory.
EDIT: the next layer of confusion is that modules (which sort of are a secret-sauce of nix and so naturally you will want to use them a bunch) are specific implementation details of multiple subsystems in nix. Meaning, nixOS and home-manager and nix-darwin all have "modules" but they are not compatible. Each has its own "idea" of what modules are and nix itself doesnt provide this natively. That means things get a little more complicated/involved when you use those ecosystems together. Its not too bad but it is annoying.
> Do you install a package or a service
A package is like the raw software installation. So eg the bash package is just a wrapper around building bash from source (theoretically from source at least... you can also define a package as being a binary distribution as long as you specify the content hash). The service (actually the _module_) is for everything else around software installation. So, eg, the bash _package_ will build bash and have it available for you to put in your path but the bash _module_ might also configure your .bashrc and set it up as your users shell. It would also generally refer to the bash _package_ so you can do all that plumbing but specify a certain version of bash you want to use.
Another common example: a plex package would again build the plex software but a plex module would perhaps create a plex user, setup systemd units, open your firewall, and create a media directory.
EDIT: the next layer of confusion is that modules (which sort of are a secret-sauce of nix and so naturally you will want to use them a bunch) are specific implementation details of multiple subsystems in nix. Meaning, nixOS and home-manager and nix-darwin all have "modules" but they are not compatible. Each has its own "idea" of what modules are and nix itself doesnt provide this natively. That means things get a little more complicated/involved when you use those ecosystems together. Its not too bad but it is annoying.