Definition:  Linux Kernel is the essential part of Linux, responsible for resource allocation, low-level hardware interfaces, security, simple communications, and basic file system management.

Good definitions. One thing to note as well, is this: Linux is the kernel, and the kernel only. All other parts of the whole as we know it (Suse, Gentoo, Mandriva) make up GNU software. The entire operating system as we know it is called GNU/Linux, not just Linux, but the general population knows what you mean when you say "Linux". Read here for an explanation and a bit of history: http://www.gnu.org/

Richard Stallman, the founder of GNU, and Linus Torvalds, the writer of the Linux kernel, together make that wonderful OS we all know and love.