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Wikipedia defines a "Kernel" as:

The central module of an operating system. It is the part of the operating system that loads first, and it remains in main memory. Because it stays in memory, it is important for the kernel to be as small as possible while still providing all the essential services required by other parts of the operating system and applications. Typically, the kernel is responsible for memory management, process and task management, and disk management.

which I don't think is a very good definition. A kernel in my mind is something which manages the computers resources (memory, disk, processor time etc) and abstracts the hardware.

fortune(6) defines a Kernel as
kernel, n.
a part of an operating system that preserves the medieval traditions of sorcery and the black art.

The marketing division of the Sirius Cybernetics Corporation defines a Kernel as

Your plastic pal that's fun to be with.

Anyone got anything that they think they should add to this definition?

Linux is an example of a MonolithicKernel, while TheHurd is an example of a MicroKernel

Remember, you can't boot a kernel by itself, hence the ReligiousWar of RMS vs common usage. Attempting to boot "Linux" would result in something along the lines of "Kernel Panic: Cannot find init". Trying to boot what RMS calls GNU/Linux would result in the OS we know and love.