A kernel is not a synonym for a function. As Zhen Lin said, it is a property of of a function.
I will try to explain the kernel of a map in linear algebra. Suppose we have a linear map $L:V\rightarrow W$. Then the kernel of $L$ consists of all vectors in $V$ which are mapped to zero. $0$ is always mapped to zero, thus the kernel is never empty. It is easy to prove that the kernel of $L$ is always a linear subspace of $V$. If $v,w\in V$ are vectors in the kernel of $L$, that is $Lv=Lw=0$, then $L(\lambda v+\mu w)=\lambda Lv+\mu L w=\lambda 0+\mu 0=0$. Thus any linear combination of vectors in the kernel is an element of the kernel. Hence it is a linear subspace of $V$.
This idea generalizes to a lot of other mathematical constructions (e.g. groups), this was what Zhen Lin was talking about.