<operating system> The essential part of Unix or other operating systems, responsible for resource allocation, low-level hardware interfaces, security etc. See also MicroKernel.
<language> An essential subset of a programming language, in terms of which other constructs are (or could be) defined. Also known as a core language.
The first sense is the most common.
Another sense is related to ImageProcessing?: a kernel is a small matrix used in convolution and various other operations.
The term is also sometimes used in reference to the "core" of some application.