/tmp and /private/tmp are the same folder (/tmp is a soft link to /private/tmp). It's the same for /var and /private/var.
There is no such thing as /var/private/tmp (I assume that was a typo).
/tmp and /var/tmp are used as temporary scratchpad space by various applications. Based on the BSD Unix directory structure (which OS X mostly follows), /tmp is used for temporary files that should not survive a reboot and /var/tmp for temporary files that should survive a reboot (although this is a bit flexible). They often contain things like launchd sockets.
/var/folders is often used for temporary downloads (eg, appliactions that update by downloading new versions of themselves often download the dmg files to a subdirectory of /var/foldersand extract them there before running their own installer).
"path to..." comes from the intersection between the OS X user domains management system and the fielsystem; it is highly context dependent and arcane.