The Arch User Repository (AUR) is a collection of user-submitted package description files (
PKGBUILD files). These description files can be used to create package archive files (
PKGBUILD files can be used by an AUR helper (e.g.
aurman, Pamac) or manually built using
makepkg. The resulting package archive file can then be installed in the normal way.
All content on the AUR is uploaded by ordinary users and very little checking of their content is done - it is up to you to verify the content is safe to use.
The AUR is not an official part of Manjaro. Manjaro systems can use the AUR but the AUR is designed for Arch, not for Manjaro.
AUR packages are targeted at Arch stable (Manjaro unstable) so may have issues when used with the older packages in Manjaro stable.
The best (most detailed and accurate) source of information about the AUR is the Arch wiki page: https://wiki.archlinux.org/index.php/Arch_User_Repository
Please read it - especially the AUR FAQ section. There's no need to repeat it all here.