The kernel is a piece of software that, roughly speaking, provides a layer between the hardware and the application programs running on a computer. In a strict, computer-science sense, the term 'Linux' refers only to the kernel - the bit that Linus Torvalds wrote in the early 90s.
The kernel of UNIX is the hub of the operating system: it allocates time and memory to programs and handles the filestore and communications in response to system calls.
As an illustration of the way that the shell and the kernel work together, suppose a user types rm myfile (which has the effect of removing the file myfile). The shell searches the filestore for the file containing the program rm, and then requests the kernel, through system calls, to execute the program rm on myfile. When the process rm myfile has finished running, the shell then returns the UNIX prompt % to the user, indicating that it is waiting for further commands.