Here's a geekier take on it.

An octave is an exact doubling of pitch, a.k.a. frequency. A above middle C is canonically defined as 440 Hz1 (two octaves above the A string on a guitar which is 110 Hz). What that "440 Hz" means in plain English2 is that the thing producing the sound is making four hundred and forty complete back-and-forth motions per second3, and pushing the air back and forth at the same rate.

Within an octave, the frequency of each note (semitone, I should say; we're using both black and white keys) is greater than the next lower one by a factor of 1.0594631 (4), the 12th root of 2. Why? Because that way, you double your frequency every twelve semitones, and the pitches you get happen to be at useful ratios from each other. Two pitches will sound nice together5 if the ratio of their frequencies is relatively simple: An octave (A + A, 2:1) sounds very pleasant, or "consonant"; a "perfect fifth" (A + E, 2:3) sounds nice, and a perfect fourth (A + D, 3:4) also sounds nice. Weird or "dissonant" ones like the augmented fourth (A + D#, 32:45) sound like crap, but they can be useful as flavoring: You wouldn't eat a handful of salt, but a little bit is a good thing.

The important thing here is that the ratio between two notes in one octave (say, A 110 and E ~164.8) will be the same as the ratio between the same two notes in a different octave (like A 440 and E ~659.3). And if A 110 sounds good with E ~164.8, you can safely assume that it'll sound good with E ~659.3 because that's the way fractions work.

I've left out some of the most interesting stuff, like harmonics, but this is probably enough for one day. Anyhow I'm not gonna try to explain harmonics without illustrations. Maybe when I hit level 6 I'll put a bitmap with some waveforms in my home node. Aren't you excited?!

1 Hz == Herz == cycles per second

2 There's no such thing.

3 This assumes that you've got it tuned perfectly, and in the real world that never quite happens, but it should be reasonably close. If it's a stringed instrument and you tune it much too high, you'll wreck the thing. The crucial thing is to agree with the other strings on the same instrument, and with any other instruments you may be playing along with.

4 This is tempered intonation. The first volunteer to explain just intonation vs. tempered intonation gets a cupcake and my undying admiration. In brief, tempered intonation is a pragmatic fudge factor thing: The ratios between notes are all made to come out slightly wrong, instead of some being perfect and others being lousy.