Dolby Noise Reduction (Dolby NR) is a system for encoding and decoding analogue audio signals on magnetic tape in order to minimise tape hiss.
Any analogue tape recording contains a constant background noise. In practical terms, music you record onto a compact cassette is spoiled by a constant underlying hiss.
The 'volume' of the hiss on the tape is constant. (It's not really 'volume' until you play it back but it's easier to explain this way). It's not related to the volume of the music you record on the tape. So, if you record the music very loudly, then the hiss is comparatively quiet. Therefore, it is best to record at as high a level as possible. You are limited, however, because the sound will become distorted if you attempt to record above a certain volume.
This is why a half-decent cassette deck has a control to allow you to adjust the input signal - you want to get it as high as possible, but not too high.
However, this is not enough to remove the hiss - merely minimise it. Enter Dolby Noise Reduction.
(I know it seems odd to discuss decoding before encoding. Trust me.)
All of the tape hiss can be judged to be within a certain band of frequencies. When NR decoding is engaged, it cuts the volume of all sounds within that frequency by about the typical volume of the tape hiss, thus removing the hiss from the output signal. Techies know this as a "low-pass filter" (although, note that it does not cut out the high-frequency sounds altogether; it just reduces their voume).
There is of course a problem with this - all the musical sounds within this frequency will be cut in volume as well, because the NR decoding circuit can't tell the difference. This tends to cut out the percussion and generally deaden the music. Not so good...
...which is why the signal must be NR encoded when you make the recording. The encoding is a simple matter of boosting the recording volume of all those frequencies that you're going to reduce in the decoding stage. This leaves the music back where it started, but without the nasty hiss. Hooray!
Dolby NR comes in three flavours - B, C, and S. (Once upon a time, there was Dolby A, now obsolete). Dolby B is the version most commonly found, present on anything from the higher-end walkmans upwards. More expensive tape decks usually also have Dolby C, and more recently the higher end ones have Dolby S as well.
They all employ increasingly sophisticated models of exactly which frequencies to mess around with, but work on the same basic principle. Audiophiles don't all agree with Dolby's assertions about which sounds better than which - some rate B better than C. Anyway, for general use, it makes most sense to record tapes in B, because there are so many more machines out there that can decode it properly.
Because the encoding process is not terribly dramatic, a tape that is encoded with NR can still be played back on a system that doesn't have the decoding circuit. It won't sound quite right - all the percussion will be too loud and it will have a bright, hard edge to it, but on the other hand if you're using a cheap nasty system that doesn't even have Dolby then that's the least of your problems.
Similarly, you can play back tapes recorded with one flavour of NR and use another to decode it - it won't sound as good as it's supposed to, but you'll hear the music.
When recording a tape, use the same type of NR that will be available on the machine you will use to play back the tape. If there's a choice, S is supposed to be best, followed by C, followed by B.
When playing back a tape, use the same type of NR that was used to record the tape. (For commercially produced tapes, this is usually Dolby B NR). Using NR to play back a non-NR tape will cut out a chunk of the music along with the hiss, so don't do that.