A photonic crystal is an artificially manufactured crystal with periodic structure such that some colors of light are allowed to propagate in the crystal while others can not. This phenomenon is known as a photonic band gap as an analogy to the electrical band gap in semiconductor materials.
The reason some colors can not propagate has to do with the ideas of reflection of light in the crystal. Although some dislike structures that are only periodic in one dimension being called photonic crystals (since the 1 dimensional variety has been around for much longer under other names such as bragg stacks, multilayer mirrors), the idea is the same and it's easier to illustrate.
I couldn't get my ascii art to work so try to use your imagination. Imagine a picture of a mirror with a multiple layers of coatings. Light enters from the left toward the face of the mirror. The critical thing to understand is that light exhibits contructive and destructive interference. When the light enters the material, some of it reflects at the first interface, some is transmitted. Each time the light encounters an interface, some is reflected, some is transmitted. If the thickness of the layers is a specific distance, the light that is reflected from interface 1 and that from interface 2 will be out of phase with one another and destructively interfere. If this is the case, light is 'discouraged' from reflecting back and will all be transmitted. Other colors with different wavelengths will not have the same destructive interference so they will not be completely transmitted or reflected. In reality, you usually do the opposite and make the phase of the light travel such that the phase will exactly constructively interfere making a very effecient mirror for a certian wavelength.
It's easy to see that this mirror will only work if the light is normal to the surface of the mirror. If the light strikes the glass at an angle, the distance traveled by the light and it's reflections will no longer be such that the phase matches up anymore (think of the distance you walk to cross the street if you go straight across as opposed to if you go diagonal down the street at the same time). Photonic crystals in two and three dimensions use the same ideas to build more complex structures so that the incoming angle doesn't matter: some colors can never propagage.
There is a frenzy of research going on in the area of photonic crystals in an effort to use photonic crystals to realize photonic computing because the photonic band gap is seen as an analog to the semiconductor band gap which is largely responsible for the information revolution.