Let's start at the very beginning. Everyone knows the three primary colours are red, blue and yellow. Secondary colours are made by mixing those, and those include purple, orange and green. Tertiary colours are created when a primary and secondary colour are mixed, such as blue and green making teal. Have a look at the diagram below to see what I mean, although I'm sure this is all old news.
Now here is the theory part that gets fun. Complementary colours are the ones across the diagram from each other. Using these colours next to each other makes them appear brighter, creates the biggest contrast, and can be very sharp. For a costuming example, if you have one piece that may be smaller but you really want to accentuate it, put its complementary colour underneath or next to it.
Next we'll talk about analagous colour schemes. Analagous colours are those next to each other on the colour wheel. Blue, teal, and green, for example. These colours run smoothly together, without much contrast or standing out.
Analagous colour schemes can also mean a variety of shades of the same hue. This can be very aesthetically pleasing.
Here is a perfect example from Flickr of these colour schemes. The dancer has on mostly shades of purple and some red, so an analagous colour scheme - but then there is also the gold colour in the belt which complements the purple, and the green hues which complement the red. All of the colours flow and improve the aesthetic of the others, creating a beautifully planned costume.