I suspect Mint might just not have anything like the AUR.

AUR stands for Arch User Repository, and it's a place where anybody can create a package. But those packages aren't going into a regular repository, instead they're kept as build scripts, simple code that describes how to make a package.

This is useful for two reasons - it allows users to share packages that aren't making it into the official repositories (because not everything will, there's just too much stuff out there), but it can also have things which can't go into the repos due to licensing (because the AUR doesn't distribute the software, just instructions on how to automatically get it)

There's no official utility to install packages from the AUR - you have to find a package you want on the site, clone the repository, and run makepkg to build and install it. And for updates you have to pull changes and rebuild it manually. And you're supposed to check yourself to make sure what you're installing is safe. But there are popular unofficial utilities that are intended to replace Arch's built-in package management, automatically finding packages both in the trusted repositories and the untrusted AUR, with no separation.