From LXF Wiki
The core of an operating system, always present and running unlike the rest of an operating system. Kernels handle task switching and processes, memory allocation, and such. In GNU/Linux, the kernel is Linux, and the latest versions of the kernel can be downloaded from www.kernel.org.
Kernels are implemented in different styles. Monolithic kernels are run completely in supervisor (aka kernel) mode, but applications are in user mode. Common examples of monolithic kernels are the Linux kernel, MS-DOS, MikeOS, DOS-based Windows, and Pre-OS 9 Mac OS kernels. Monolithic kernels sometimes have loadable modules, like Linux.
Microkernels run only the most important core services in kernel mode, with all of the drivers, modules, etc. running in user mode with the applications. The Mach kernel is a microkernel.
Hybrid kernels combine the best and worst of both kernels by handling important services in kernel mode and applications/less important services in user mode. The Windows NT kernel and Plan Nine are hybrid kernels.
TODO: Add more about what kernels do.