Chromatography is a technique to separate out the various compounds in a mixture. For example, in a forensic case, if it was suspected that a glass of water was contaminated with a poison, a lab could take what was left in the glass and use chromatography to separate the poison from the water.
There are several different types of chromatography - thin layer (paper), high performance liquid chromatography (column), and gas chromatography. All three work using the same fundamental principal - If you are trying to separate substance 'a' from substance 'b', substance 'a' and substance 'b' will be attracted differently to a third substance 'c'. You can use this difference in attraction to separate 'a' from 'b'.
In each of the types of chromatography you have a 'stationary phase' and a 'mobile phase' that act to attract the substances you want to separate. The stationary phase will have one property while the mobile phase will have the opposite property. For example, the stationary phase might be hydrophobic while the mobile phase is hydrophilic. If substance 'a' is hydrophobic, it will be attracted to the stationary phase (in this example) and if substance 'b' is hydrophilic, it will be attracted to the mobile phase. This will separate the two compounds and allow you to collect them individually to identify them using other techniques such as mass spectrometry.
To learn more about this process and to see it in action, explore the links below.
Introduction to Chromatography - view this animation to see the basic concept of chromatography using thin layer and gas chromatography as examples.
Thin Layer Interactive - here you can interactively affect the substances in thin layer chromatography.
Column/Liquid Chromatography - watch this YouTube video to see how liquid/column chromatography works. hint: same idea, but different method
Column Chromatography Animation - here is an animation in case the above YouTube video doesn't work.