Convolution and Cross Correlation on 2D Image
If as you said you understand well the 1-D convolution/cross-correlation functioning (the Wikipedia first graph explains it in a clear way), the 2-D version is very similar!

This website explains 2-D convolution in a simple way with clear indicies and examples. In a nutshell, the kernel has to be flipped for convolution and this means your kernel example would become 
[M L; K J] (sorry for the formatting). Then you multiply elementwise and sum the results, as you would do in the 1-D case.

Cross-correlation functioning is the same without flipping the kernel.