The AUR is the Arch User Repository. All it is is User uploaded software packages with a script that Arch Linux and its many derivatives recognize and know how to utilize to install a piece of software and the necessary libraries/dependencies on your system.

It is similar to Debian based systems when you install software that’s not in the officially repos by appending an unofficial mirror to :

/etc/sources.list.d

Take installing Mullvad VPN on Debian for example. It’s not in the official repos, so you have to tell apt where to go get it.

Paru and Yay are what are known as AUR Helpers. All they are doing is automating the update process of the packages you installed from the AUR, which normally you’d have to update one by one manually. They also can help you easily search the AUR from the command line.

In essence, they are wrappers around pacman and makepkg.

Flatpak is different in that it is an OS agnostic package manager that sandboxes applications away from the main OS and essentially downloads/installs all its libraries and dependencies into ~/.local/share/flatpak instead of /usr/lib, though this is a vast oversimplification.

Very basically, paru/yay says “We install stuff on Arch and Arch based distros from unofficial, user maintained sources and keep them up to date so you don’t have to update them one by one once installed. When possible, you should probably default to just installing with pacman and using the official repos though.”

Very basically Flatpak says “I don’t care if I’m run on Debian, Arch, Gentoo, whatever, I’m bringing all my system libraries with me cuz I don’t know what version of what is on here, and I just need this app to run right the fuck now. So even though it’s heavier and less efficient, here’s plumbing and the kitchen sink so you have running water right fucking now. BTW, you probably shouldn’t run anything installed with me as root.”

This is a very oversimplified explanation, but hopefully that helps clear things up for you.