A kernel is the core part of the operating system that defines an API for applications programs (including some system software) and an interface to device drivers. On some systems the kernel may be called "nucleus" or "hardcore".
Device drivers, including also computer BIOS and device firmware, provide basic functionality to operate and control the hardware connected to or built into the computer.
In contrast to system software, software that allows users to do things like create text documents, play games, listen to music, or web browsers to surf the web are called application software. The line where the distinction should be drawn isn't always clear. Most operating systems bundle such software. Such software is not considered system software when it can be uninstalled without affecting the functioning of other software. Exceptions could be e.g. web browsers such as Internet Explorer where Microsoft argued in court that it was system software that could not be uninstalled. Later examples are Chrome OS and Firefox OS where the browser functions as the only user interface and the only way to run programs (and other web browser can not be installed in their place), then they can well be argued to be (part of) the operating system and then system software.