What is single user mode useful for? I don't quite understand on the Internet I couldn't really find anything useful. What is the individual user there for exactly? The terminal that is used in single user mode can also be used in the recovery partition?
Single user mode is there in order to be able to diagnose and fix problems with the system. It is commonly used if there's an error with the system that prevents it from properly booting normally into multiuser mode (i.e. the default mode).
The single user mode allows advanced users to examine the file system, change settings, and otherwise work with the system and installed programs. In theory, you can run any command line program here - so if needed, you could for example copy all your data to an external drive in order to save files from a non-working system.
The command line interface when booted in Recovery mode is similar, but actually not the same. In single user mode you're booting your actual operating system that you would normally start up. In Recovery mode you're booting a separate &quot;known good&quot; system that might or might not be the same as your primary system (i.e. it could be an older or newer version for example). In Recovery mode you also wouldn't have immediate access to programs that you have installed yourself. You can mount the primary drive and access your programs that way, but it is not exactly the same.
Note also that single user mode predates the Recovery partition by quite a lot as it has been there since the original OS X Cheetah release in 2001, whereas the Recovery partition was introduced with OS X Lion in 2011.