This is actually very closely related to what I do for a living (something called glow discharge that is used in chemistry and physics).
Basically, a tube is filled with one of the so-called noble gases. This is sometimes neon, but also often argon. There are two pieces of metal (electrodes) at opposite ends of the tube. Very high voltage electricity (thousands of volts) is applied to the electrodes. This potential breaks some of the neon (or argon) atoms into positive ions and electrons. The voltage (which is switched quickly so that it is first going in one direction, than the other, than the first direction, and so on) first pulls the ions in one direction, than the other. As they collide with gas atoms in the tube, they give some of their energy to these atoms. Some of the energy is used to energize electrons within these atoms. In order for the electrons to relax, they must release their energy. The main way that they release this energy is in the form of light.
You don't usually see this light directly, however. The light you see is from something called a phosphor, which is any compound that phosphoresces. Phosphors absorb this light and then emit light of a different color (whcih is how you choose the color of the neon light).
You might be interested in
this topic about plasma TVs, because they work very similarly.
