This is going to be a rather lengthy post, full of analogies, basics in physics, and definitions, so please bear with me. I will have to assume a certain amount of math and physics knowledge to keep this from going on forever tho. If I gloss over anything you would like explained in more detail, just let me know.

First off, it needs to be assumed that sound is nothing more than vibrations of air. Something causes the air to vibrate at a certain range of speed and the ear will detect this and your brain will convert it to sound. The faster the vibration, the higher the pitch of the tone.

A frequency is how often something happens on a regular basis. A clock ticks every 1 second. One second is it's frequency. Hertz is a measurement of how often something happens in one second. 440 hertz means that whatever object this is (guitar string, electric signal, air column) vibrates 440 times in one second.

Imagine pushing a child in a swing. Each time she (I have two daughters, so I always swing "she"s) swings back, you give just a little nudge. She swings forward again; comes back again. You give another little push, she goes a little higher. The distance that she moves forward (and back) from the starting point is called the amplitude and is considered higher the more distance is covered. The timing of your push must match the instant the child reaches the high spot of the swing. If you try to push when she is still swinging back, what happens? The first thought is that someone gets knocked on their tuckus, but the answer I had in mind is that the swinging gets slowed down, or dampened. When two actions (vibrations) coincide in such a manner that they enhance one another (the child swings higher) they are said to resonate. The time period from when you pushed, thru the swing forward, the swing back, to the point you are just ready to push again (one cycle) is called the natural resonance frequency.

Everything has a natural resonance frequency (NRF). Hum a note, any note. Two days later, hum a note. Chances are that you will hum the same note. That is the NRF of your vocal chords. My NRF is a slightly sharp F#. Is there a note that you sing that makes your chest vibrate or feel funny? That note is the NRF of your chest wall.

When you take a stemmed glass, wet your finger, and trace it along the edge of the glass, it will produce a tone. What is happening is that your finger sticks to the glass and actually pulls it a little bit. Then the two surfaces un-stick and the glass rebounds. They stick, pull, release, rebound, setting up a vibration. When the vibration gets enhanced enough, you hear the tone. (BTW, both your finger and the glass must be oil and soap free. They need to be clean so that the friction can take effect.) That tone is the glass' NRF.

You need to use a stemmed glass because the stem isolates the bowl of the glass from the surface it is sitting on, allowing it to vibrate more freely. You can do this with a tumbler type glass, but it is much more difficult to get the tone because the contact with the table dampens the vibrations of the sides of the glass.

The more pure a substance is, the stronger it will react to it's NRF. If you have something made up of a number of different things, each with its own NRF, they will work to dampen each other. There will still be a NRF for the unit as a whole, but it won't resonate as strongly as a pure substance.

Vibrations of an object generally move in a lateral direction from the length of the object. For example, a guitar string vibrates back and forth across the face of the instrument, which is across the direction of the length of the string (running nut to bridge). The mouth of a glass has the length bent into a circle, so the vibrations must move crossways from this direction. Since the glass won't stretch, you will get the glass mouth deforming into an ellipse, first running side to side, then switching to front to back. The alternation of these two states is your vibration.

While glass can bend a small amount, it will break if bent very far. So, take your stemmed glass. Start playing the exact tone (no overtones or harmonics) that is the NRF of this glass. The glass starts to vibrate. Continue playing the tone; the glass vibrates with larger amplitude, deforming the shape of the mouth of the glass. Add volume to the tone (a harder push), the amplitude increases again, and you will reach a point where the vibrations in the glass cause the glass to bend more than the materials are able, and the glass will shatter.

Crystal, while not only being a very pure glass, also contains a high percentage of lead (which, BTW, is why you shouldn't store wine in a crystal decanter - the wine leaches the lead and the next time you drink the wine you're drinking all that lead. Once isn't enough to cause much harm, but if you drink it often enough...) The lead makes the glass stiffer, but more brittle. (Don't ask me how soft lead can make glass stiffer - it's a chemical reaction thing, and I don't profess to understand it.) Crystal glasses also tend to have narrower stems, thus reducing the dampening effect of the stem. So crystal works better than "regular" glass because it reacts strongly to it's NRF, is freer to vibrate, and is more brittle. Overtones and weak NRF reactions due to impurities in glass will generally keep a regular glass from breaking, but given enough volume and duration of the correct tone, theoretically, it can be done. (Overtones will cause the mouth of the glass to vibrate in a combination of ellipse, trianglar, square, etc. shapes, each dampening the others.)

Olafur - The best way to find the NRF of a glass is the old finger on the rim trick. For other things, start generating a loud tone of low pitch and gradually increase the pitch, checking the object for resonant vibrations. When it is vibrating at it's strongest, the pitch being played is that objects NRF.
Yes, by pure tone I mean Sine wave free of overtones and harmonics - they will tend to dampen the resonance. (BTW, Sinus wave brought to mind "the sound made when you blow your nose on a regular basis." *G*)

Joe - "Glass" is a very generic term. It is very similar to "cake" in that manner. How many different kinds of cake are there? Depending on the intended use of the glass, it can have a variety of recipes, include different amounts of impurities, etc. Ever notice how, when viewed edge on, some glass has a blue tint, other seems green, and other seem gray? This is due to the ingredients in the glass. "Crystal," while still a generic term, is much more specific - akin to saying "Pineapple upside-down cake." It's still a cake, but more specific.

Jeri - you got it exactly. And make it sound so much simpler than I do.