Separation by partitioning between phases
Chromatography exploits the fact that molecules stick to different materials with different strengths. A sample is loaded onto a stationary phase (a solid, a liquid coating a solid, or a gel). A mobile phase (liquid or gas) flows through, carrying molecules at rates that depend on how much they prefer to stay in the stationary vs. mobile phase. Molecules that stick weakly move fast; those that stick strongly move slowly. This difference in movement separates the mixture.
Gas chromatography (GC) uses a gas as mobile phase, typically a heated column packed with a liquid coating or polymer. Molecules with lower boiling points and lower affinity for the stationary phase elute first. It is fast, quantitative, and excellent for volatile, heat-stable compounds.
Liquid-phase and specialty methods
HPLC (high-performance liquid chromatography) uses a liquid mobile phase under pressure, giving much finer control and applicability to larger, more polar molecules that would decompose in a GC column. TLC (thin-layer chromatography) is the DIY version: a thin layer of adsorbent on glass, a drop of sample at the bottom, solvent flows up. It is cheap and fast for rough separation.
Ion-exchange chromatography separates by charge: the stationary phase is functionalized with charged groups that attract oppositely charged molecules. Proteins, amino acids, and nucleic acids are routinely purified this way. Size-exclusion (gel filtration) separates by size: large molecules elute first because they cannot enter the pores of the gel and skip through the column faster. Each method solves a different separation problem.