understand root and su As I understand to "Root" a device, means to gain privilege to do all sort of things with the device, without restrictions, which probably means "root privilege".

Isn't that what "su" utility provide , i.e. root privilege ? I mean is all we need is to run su in order to have root privilege or are there any additional things, that should be done ?
Yes, the su utility is all you need, but the problem is putting the su binary (basically the .exe of Linux and Linux based-OS's) onto the phone. In order to put su onto the phone, you need root in the first place. su is usually added to the phone by unlocking the bootloader, which allows it to run a recovery which can flash su onto the phone.