Terminology
Paper chromatography (PC) is a lot like thin-layer chromatography (TLC), a quick and convenient test chemists often use in the lab. As its name implies, however, PC relies on a sheet of filter paper to effect the separation. The paper starts out with its lower tip suspended in a layer of solvent like water or rubbing alcohol. As the solvent climbs up the paper by capillary action, it carries the molecules in your sample with it, and differences in molecular structure cause some molecules to migrate more slowly than others.
Mixtures
You can separate a lot of mixtures. Colorful mixtures are the most fun because you can watch the colors separate. Food coloring and sharpie markers are favorite choices for introductory labs. Dyes from M&Ms or Skittles are a more adventurous choice, although they take more work because you need to extract the dyes from the candy, soak them up onto some yarn, then release them from the yarn to ensure you're getting only dye and not contaminants.
Setup
Once you've got your materials, you can start your setup for the experiment. Take a rectangular sheet of filter paper big enough to fit inside a beaker without bending or twisting it. Draw a line about ~1 - 2 cm or so above the bottom of the sheet, attach a paper clip with a string looped through it to the top of your paper and hang it inside the beaker.
The Experiment
Pour a little solvent into the bottom -- a layer maybe about half a centimeter thick or so -- and cap the beaker with a watch glass. Verify that the bottom of the paper isn't touching the solvent, and leave it for half an hour so the air inside becomes saturated with vapor. Remove your chromatography paper, and spot a tiny dab of the sample onto the line you drew, which will serve as the starting line. Placing the paper back in the beaker with the bottom of the paper in the solvent will allow the solvent to soak up into the paper. See that the line with the sample(s) on it is above the solvent layer, or your samples will dissolve and you won't get any results. Place the watch glass back on the beaker, and watch the sample ascend the paper.
Considerations
Once you've learned the technique, try experimenting with different concentrations of solvent. You could use rubbing alcohol in water, for example, but vary the concentration of rubbing alcohol. You can also make the experiment more quantitative by drawing a line to mark the farthest position reached by the solvent after you remove the paper from the beaker. Measure the distance from the starting line to the point reached by each compound, then divide it by the distance travelled by the starting front. This calculation will give you a number called the retardation factor (Rf) so you can compare different compounds.