There are lots of different types, and I also find them confusing – I have to check with my mum (a science teacher) about how they work! How about we go with simple paper chromatography?
It works on the idea that you can separate solutes from a mixture using solvents based on how heavy/small the molecules or particles are. The tinier/lighter it is the higher up the paper it travels because it is more soluble. Using this method people can identify the different things in a mixture by making an Rf value, that is the ratio of the distance of solute divided by distance solvent moves, which is always less than 1. This number is then compared with a data booklet of other known Rf values so you can find out what your numbers are and therefore work out what is in the mixture! They used to use this method to work out whether things had been forged by looking at the contents of ink, to see if it was the same as the ink on manuscripts of the same age or written using the same pen for example.