Signal to Noise Ratio

I was comparing the specs on a couple of cassette decks I own. I have always known that the higher the S/N the better. But what exactly does a greater S/N mean?
I'll take a stab at this. Let's say the signal is only the charged particles (the music on the tape) and the noise is the scraping of the tape over the heads. Everything else in environmental, mechanical, electrical and radio contact with the signal is making noise too but forget about it for simplicity. Ok now imagine making salad dressing. Just olive oil (levels of charged particles) and vinegar (levels of noise). One rides atop the other. In real life they're mixed up, but in a visual sense the signal dominates the noise and rides above it. Ok? Now let's say the whole bottle is filled with dressing to a level called 0dB. How far down (from loud to soft) into the oil can you go until you hit the vinegar? The signal-to-noise ratio tells you. For example, if the ratio is 60dB, you'll hear all the signals that occur from 0dB (the loudest reference) all the way down to 60dB less loud than that. Softer than that, the noise will overcome the signal and obscure its perception. Good luck!
That is slightly different from my understanding. The term Signal to Noise Ratio is just that, a ratio. My understanding is that the idea is a 100S/N ratio means that for every 100 dB's of signal, you get 1dB of noise. Thus you want as much signal as possible for every bit of noise.
Most of the info already posted is essentially correct, save a few points. S/N ratio is a bit of misnomer, although it obviously expresses a relationship between the input audio signal and the noise inherent in the system (the electronics primarily, but "dirty" electricity can also contribute). To be a bit more precise, the noise in the system is best expressed as a "minus" figure below the signal. Hence, a S/N ratio of -96 db (pretty typical for CD's) means that the noise level is 96 db below the audio signal. The higher the S/N ratio number, (e.g., -100db), the lower the noise level in relation to the signal level. Some of the newer power amps, such as Bryston's 4B-ST, now have S/N ratios around -116 to -118 db, so they are actually quieter than CD's. The S/N ratio really becomes a factor when you are listening to very quiet music, since this is the time when it is most likely for the noise to be audible. On loud musical passages, such as heavy metal rock, the S/N ratio is essentially unimportant. Likewise, very weak audio signals, such as those from a low output phono cartridge (which can be less than 1.0 mv), may be affected by the noise. Hence, S/N ratio can be a significant factor in reproducing sound from LP's. Hope this helps.
SDCampbell, in your example you say the noise level is 96Db below the signal yet claim that at low signal the noise becomes more apparent. Of course the latter is correct. I merely point out that a signal to noise ratio is a measurement in relation to a reference level of 0 dB, the highest level used in the measurement. In other words I do not believe the ratio expresses an absolute deviation from whatever signal is present, but is only in relation to the highest signal used. Hey, maybe it's only me :-)
Rockvirgo - you are correct. S/N is the measure of noise below 0db level, whether the music is at that level or not. Typically music rarely plays at 0db. In an analog system such as a tape deck, 0db is already into some distortion. At +3db most tape recorders are well into distortion. For digital recording 0db is totally pegged ie its the maximum output for the 16bits being put into a word on the CD. Now most music will be layed down on the recording medium below these maximum values. The music you listen to might have a dynamic range of 50db and the loudest moment in the music recorded at -5db. This means to softest music is at -55db. But that's not to softest signal on the recording. Backgroud noise (outside street noise) musicians shuffling music, and good stuff like the ambience in the recording hall all come in below (hopefully) the softest music. Now if the device doing the recording has a S/N ratio of 100 db then there is 45db S/N ratio for the softest passage and 95 db S/N for the loudest. This is the S/N for the recording, not the equipment. It's that 45 db of floor that really makes the recording sound good. You pick up more "air" and "ambiance" in your listening experience. The recording process can also shift the balance by micing near or far from the instruments. A close micing will allow less ambience to be recorded because the ratio of instrument to hall ambience shifts many db. It does however give you more detail in the instruments near field sound that includes inside-the-instrument harmonics and overtones as well as instrument and musician noice (value clatter on a horn or the musicians snorfing - ever heard a violinist or pianist humming along with his own playing ala Glen Gould)