Paper chromatography
Chromatography was first used to separate plant pigments such as chlorophyll and carotenes, which is how it got its name, because these are colored molecules.
Take a piece of thick paper (cellulose), spot a sample of spinach extract on it, and then place the paper in a jar of solvent (spot above the liquid). Capillary action moves the solvent up the paper, and if the components of the sample are soluble in the solvent, they will move up as it rises. The components move at different rates, so they get separated.
Here is an idealized result:
A typical solvent would be a mixture of petroleum ether, acetone, and water (3:1:1). Various protocols are available on the web. [link] Luckily petroleum ether is not really an ether so it doesn't form explosive products, although it is flammable.
DEAE
One advance was to manufacture derivatized cellulose where other chemical groups are attached. Cellulose is just glucose units with a particular linkage.
Two very common substitutions are DEAE (diethylaminoethyl-cellulose), which is positively charged around pH 7.
Another advance is to pack the material into a column and use gravity or a small pump to move the solvent through the column under moderate pressure.
Here's an example from one of my old lectures where extracts of E. coli are fractionated on DEAE-cellulose to separate the different DNA polymerase activities. On the x-axis is the fraction number, the first material to come out (elute) from the column is on the left, last on the right. The y-axis is the DNA polymerase activity.
The polA mutant lacks the primary DNA polymerase activity (Pol I), yet it grows fairly normally. That suggests that one of the other activities is responsible for replication of the chromosome. Mutants lacking Pol I are sensitive to mutagens such as UV light, which suggests that the primary role is in repair of damaged DNA.
Another very common substrate for protein purification is phospho-cellulose, where phosphate is attached. This is negatively charged near pH 7. DNA and RNA polymerases bind pretty tightly to this.
An important modification (used for the column above) is to change the composition of the buffer in the column as time goes on. Here is a simple device for doing that, called a gradient maker.
You put say, low salt buffer in the left cylinder, and high salt buffer on the right. Open the valve between the two, and pump out from the left hand side. Mix well as you introduce high-salt into the low-salt chamber. The result is a linear gradient from low salt to high salt.
Since proteins bind more or less tightly to DEAE (and P-cell) by interactions with charged amino acids, they will "come off the column" at different salt concentrations.