Distillation - used for mixtures of liquids with different boiling points, or for a solid dissolved in a liquid.
Chromatography involves the separation of different dissolved substances as they travel through a material. The dissolved substances are separated based on their interaction with the stationary phase.
Electrophoresis Organic molecules, such as protein are placed in a gel. A voltage is applied and the molecules move through the gel because they are charged. The gel restricts the motion so that different proteins will make different amounts of progress in any given time.
A heterogeneous mixture of solid and liquid or solid and gas is usually fairly easy to separate because of the 2 different physical phases.
A mixture of two or more solids is usually separated by utilizing the different chemical or physical properties of the substances.
Chromatography works to separate a mixture because the components of a mixture distribute themselves differently when they are in contact with a "two phase system".