What is the difference between a kernel and a function? I have been looking around for this question, but all results I found only describe the definition and not the answer I seek.
Is "kernel" basically a synonym of "function"? When should be the time we should use the word "kernel" instead of "function"?

"Kernel" is an old-fashioned term for the function you use to define certain integral operators. (I assume this is the sense you mean, not the more common modern sense, which is completely different.) Like many other words in mathematics (although people generally never tell you this), it has less to do with denotation than connotation: when you use the word "kernel" you are thinking of your function in terms of integral operators. 
