I was wondering if there was any resources available online that explains what happens with something, like printf of C, that explains what\'s going on in the very low level
By definition, BIOS and kernel calls are platform-specific. What platform are you interested in? Several links to Linux-related information have already been posted.
Also note that printf may not even result in any BIOS or kernel calls, as your platform may not have a kernel or BIOS present (embedded systems are a good example of this).