The "kernel" is the main part of the OS that controls hardwares once you are past the initial post tests performed by the bios. One of the main reasons Vista is seen as more stable then XP was the method MS used for extracting various processes from the kernel and seeing them become user defined.
The big gain now seen over both XP and Vista is due primarily to the refinement now being seen in the Windows 7 kernel based on Vista's. That allows for the use of Vista device drivers on the upcoming version while seeing a drastic reduction of what is termed "bloat".
Linux and other OSs use their own type of kernel there. For both OSs the two links here will give you a good description for each.
For MS Windows, http://what-is-what.com/what_is/microsoft_windows.html
What is kernel?
The kernel is a program that constitutes the central core of a computer operating system. It has complete control over everything that occurs in the system.
A kernel can be contrasted with a shell (such as bash, csh or ksh in Unix-like operating systems), which is the outermost part of an operating system and a program that interacts with user commands. The kernel itself does not interact directly with the user, but rather interacts with the shell and other programs as well as with the hardware devices on the system, including the processor (also called the central processing unit or CPU), memory and disk drives http://www.debianhelp.co.uk/basics.htm#What_is_kernel