The core of the operating system. The kernel has complete control over everything that happens. When your computer crashes, it means the kernel has crashed. If only a single program crashes but the rest of the system remains running, then the kernel itself hasn't crashed. The kernel is designed to coordinate among the different components of the operating system, such as disk drive, networking, keyboard, and running programs.
Key point: The kernel is responsible for security, preventing one program from one user from breaking into other programs running on the same system. All systems except the older Mac and Windows do not provide this level of security.
Key point: The kernel itself does not interact with the user. For example, the word "Linux" really means just the kernel. What we see in Linux distributions is actually the kernel plus a whole bunch of UNIX-like applications built on top of it.
Contrast: There are two modes your software might be running in. Kernel mode is running within the context of the kernel itself, and describes not only the kernel but also device drivers. User mode is the context the rest of the software runs in. The key point is that when you are running in kernel mode, you've got access to the entire system and nobody can stop you. However, when you are running in user mode (especially when not logged in as root), then the kernel imposes security on your activity. This means that if you break into a normal HTTP server, you may not actually be able to break into the entire machine. However, newer versions of the Linux kernel are putting things like HTTP servers into the kernel itself (for performance reasons). An exploitable flaw in such services will allow the entire machine to be compromised.