The tar.gz and .deb files you are seeing online are both software, however the .deb files are actually compressed directories which contain all the of the software’s binaries, and are simply bundled in order to allow Ubuntu or Debian Linux distributions to open and install them with a click of a button using their “apt” package manager. OpenMandriva does not use the APT package manager, instead using one called DNF (google fedora DNF) which handles .RPM files.
As for the tar.gz files you are finding, these too are simply compressed files with the software’s code (compiled binaries usually). Unlike windows, on linux you often have to give a program (or file) permission to execute, be read, written to, etc. The difference with these tar.gz downloads is that they aren’t for use in a package manager, but rather your supposed to simply download them, extract them, and make either A: The installer contained (ending in .sh, .py, .jar, etc) or B: The program binary, executable (lookup the command chmod +x … Side note: You can also usually make a program executable through your desktop environment of choice in Linux by right clicking the file in the file browser and marking it executable).
So while you can indeed install software from a website in Linux, and sometimes might only be able to do just that (aka: software not found in the repository of your distro), it’s generally recommended to get the software from your distribution’s package repositories first, as it’s not only safer (the packages are generally vetted for stability and security), but also easier.
You can also choose to use snap packages or flatpaks (flatpak org), which are just a alternative to your distro’s repositories (packages from these locations are usually sandboxed, can be more up to date sometimes, and are self-contained so that they can be installed on any distribution you choose). But i wouldn’t worry about that right away.
Something to note: Unlike windows, OpenMandriva (or whatever Distribution your using) isn’t really the operating system, but rather a bunch of software customized and bundles with their own tools and repositories (and whatever version of the Linux kernel they bundle 5.1 for example vs 5.0 or 4.19). The “operating system” GNU/Linux is the kernel and software underneath (such as the compiler for example), which run the computer and do most of the heavy lifting. I think this is an important distinction to draw, because if something isn’t working on one distribution, switching to another can sometimes fix the issue (stuff breaks on Linux too unfortunately).
I’m more than willing to answer any further Linux questions you have in a PM. I’m far from a expert myself though.