How do Glowsticks Work? – The portable Chemistry Experiment!
Within each glowstick there are 3 chemicals:
Hydrogen Peroxide
Phenyl Oxalate Ester
Fluorescent Dye (to give colour)
The light you see emitted from a glowstick is caused by a chemical reaction between hydrogen peroxide  and  phenyl oxalate ester. For this reason, the two chemicals are held separately inside the glowstick until you ‘crack’ it.

How are the two chemicals seperated?
The glowstick is not just a single cylinder, it is actually two cylinders, one inside the other. The inside cylinder (or vial) which usually contains the Hydrogen Peroxide is made from Glass and is much easier to break than the plastic outer cylinder. The outer cylinder contains both the glass vial, and a solution of Phenyl Oxalate Ester (so you basically have one chemical floating within another).
When you ‘crack’ the glowstick, the internal glass vial breaks allowing the hydrogen peroxide to mix with the phenyl oxalate ester. This produces the emitted light!

Ok, so I understand that two chemicals mix, but why does this produce light?
(Note: This has been taken from HowStuffWorks.com as they are great at explaining things):
We saw that light sticks use energy from a chemical reaction to emit light. This chemical reaction is set off by mixing multiple chemical compounds.
Compounds are substances made up of atoms of different elements, bonded together in rigid structure. When you combine two or more compounds, the various atoms may rearrange themselves to form new compounds. Depending on the nature of the compounds, this reaction will cause either a release of energy or an absorption of energy.

Light sticks come in a variety of colors. The color of the light is determined by the chemical make-up of the fluorescent dye in the stick.

The reaction between the different compounds in a light stick causes a substantial release of energy. Just as in an incandescent light bulb, atoms in the materials are excited, causing electrons to rise to a higher energy level and then return to their normal levels. When the electrons return to their normal levels, they release energy as light. This process is called chemiluminesence.
The chemical reaction in a light stick usually involves several different steps. A typical commercial light stick holds a hydrogen peroxide solution and a solution containing a phenyl oxalate ester and a fluorescent dye. Here's the sequence of events when the two solutions are combined:

1. The hydrogen peroxide oxidizes the phenyl oxalate ester, resulting in a chemical called phenol and an unstable peroxyacid ester.
2. The unstable peroxyacid ester decomposes, resulting in additional phenol and a cyclic peroxy compound.
3. The cyclic peroxy compound decomposes to carbon dioxide.
4. This decomposition releases energy to the dye.
5. The electrons in the dye atoms jump to a higher level, then fall back down, releasing energy in the form of light.

The light stick itself is just a housing for the two solutions involved in the reaction -- essentially, it is portable chemistry experiment.

Why do you have to break glow bracelets in several places?
Glow bracelets, due to their length, contain many very small Glass Vials. So in order to make the whole stick glow brightly, all of these glass vials need to be broken which generally requires the bracelet to be bent in all places.
As a side-note: In 2007 I went to a glowstick factory in Shenzhen, China. I was given a tour of the facilities and shown how the glow sticks were made. Astonishingly, glowsticks are basically Hand Made! People working in the factory have day-jobs manually putting each Glass Vial into the bracelet. Although I did not count I would estimate there are around 50 small glass vials in each 20cm Glow-Bracelet!
You could easily check yourself by cutting one open! Just make sure you wear gloves!

A fun Classroom Activity:
Want to teach your students how they work? Why not put in a pair of gloves and cut a glowstick open to show them the separate parts and how they work? Kids love this kind of hands on stuff. Although the chemicals are non-toxic I would recommend using rubber gloves to be safe. And, like all science experiments, Ingestion should be completely avoided.