The folder called "System Folder (from Old Mac)" is your old OS 9 that is used for Classic compatibility.
You can trash it if you don't need Classic any longer.
The folder called "System" is part of OS X and contains, within its Library, vital extensions, caches, and Core Services such as the Finder, the Dock, etc.
The actual Tiger System software is the kernel, which lives a hidden life directly on the top level of your hard drive (/) as /mach_kernel and /mach.sym, with vital support files in hidden folders at the uppermost level of your hard drive called /bin, /cores, /private, and /sbin. All drives are mounted within a point in your filesystem at /Volumes/the_name_of_the_drive. These folders are hidden because you really, really don't want to touch them, nor even bother looking at them.
The /System/Library folder is one you occasionally need to see, because that's where things like printers or audio drivers are installed, but you never touch.
The /Library folder is where plugins that are to be used by all users on the machine are installed, such as Internet plug-ins, audio loops and effects plug-ins, etc.
The Library folder in your home directory performs many of the same functions, but those affect only the user in whose Library folder the plug-ins are installed.
Those folders are vital for system operation, and they need to stay exactly where they are.