There has been a great deal of good advice, but there have also been a couple of "old wive's tales" that really need to be put to rest...
the decibBel is, as stated, 1/10 of a Bel, which was a unit-less measurement introduced by the Bell System to make transmission loss easier to work with. The original unit of measure (really testing my memory here) was Mile of Acceptable Cable. No, I'm not kidding!
None of which matters... what matters is that the dB is a ratio, and specifically, a power ratio. By definition it is an rms measurement - it can not be properly used to represent peak levels in analog measurements (more on this in a moment.)
rms means root-mean-squared, another handy little mathematical too that lets us calculate the effective power in a periodic electrical (or magnetic) wave. It is equal to the DC voltage that would produce the heat (dissipated power) in a pure resistance. All if which is important if you are a measurement geek, for us the things that matter are that it is used to measure the effective power in a periodic wave, and there is very little music that can be represented by a periodic wave, but all music is made up of periodic waves.
The dB ratio is equal to 10 times the log of the ratio of two power levels. If you know Ohm's Law you can manipulate this a little bit and come up with the forumla of dB = 20 times the log of the ratio of two amplitudes or levels (or in our case, voltages.) And for the curious, the forumula for the Bel is the log of the ratio of two power levels. But if you were curious you probably already guessed as much.
dB becomes much more useful when you replace the bottom half of the ratio with a known value.
In the beginning there was dBm0 which meant the ratio of the power in a circuit with respect to 1 mW across 600 ohms. This is still quite useful in telephony and RF, but it doesn't help us much, matched power transmission went out of style in the early 1970s.
We are much more interested in dB scales that reference voltages. The first such scale used the equivalent voltage one would measure in that 1 mW signal across 600 Ohms, or approximately 0.7746Vrms. We refer to that today as 0 dBu, and there is no requirement for impedance, it is simply a voltage ratio.
It remains the basis of the "professional" audio interface. +4 dBu has become the reference level for a great deal of equipment, most often balanced inputs and outputs. That is not a requirement.
As consumer "Hi-Fi" gear grew in popularity the balanced interface was replaced with a single-ended interface, and the nominal operating level was changed from +4 dBu to -10 dBV. The dBV scale uses as its reference 1 volt. A lot easier to do the math, but it never really took off in professional circles. (Even today many broadcasters think +4 dBu is a silly standard, they prefer +8 dBu.)
Wandering off just a little bit - but there is a good reason - lets convert our nominal levels to their voltages:
+4 dBu = 1.227Vrms = 3.472Vp-p (and approximately 1.78 dBV)
-10 dBV = 0.316Vrms = 0.894Vp-p (and approximately -7.78dBu)
The two nominal levels are approximately 11 dB apart - just for grins!
OK, so what about this pesky digital audio?
All of the sudden we need to be able to think in terms of peak voltage if for no other reason that we do not wish to clip our converters. From this was born the dBFS scale, and it is slightly different because the reference is the maximum, there is no way - by definition - to go over 0 dBFS.
Which, as it turns out, really doesn't matter a whole lot either!
What matters, and what many folks have been trying to say, is that the relationship between 0 dBFS and the maximum analog voltage you expect to need is the important thing.
For example, lets say I have a microphone preamplifier that is capable of putting out +24 dBu (about 12.7Vrms or about 34 Vp-p). If we set 0 dBFS to equal +24 dBu then our nominal operating level will be about -20 dBFS if we are thinking in terms of +4 dBu analog audio.
That paragraph may trip up a lot of folks so I'm going to leave the math portion of this post for now - please feel free to offer questions, corrections, etc...
So when audio became a thing some bright folks noticed that neither a peak reading nor an RMS reading meter accurately predicted squat - you couldn't use it to manage modulation in your transmitter, and you couldn't use it as a benchmark about loudness. And that's what we care about!
So a new measurement was developed - the Volume Unit. You've seen (I hope) VU meters on audio gear. What makes a VU meter special is the ballistics, or how quickly it reacts to positive and negative going changes. It is WAY to slow to show a typical musical peak. But once you get accustomed to it you can get a pretty good sense of relative loudness.
In pro audio 0 VU = +4 dBu, in consumer gear 0 VU usually = -10 dBV. AHA, that dB stuff does matter eh?
No one that I am aware of has come up with a digital meter that mimics the ballistics of an analog meter. The Durrough meter plugin from Waves comes REALLY close. There are others too, I'm sure, but I stopped looking a long time ago and just use analog meters<G>!
The meters in most software platforms do a really good job of reporting peak levels, and for the most part they do a decent job of reporting RMS levels. They just don't rise and fall with the same timing as the old mechanical meters.
Which may well be a good thing! In the old days we didn't care much about peaks because there were about a dozen things in the audio path that would squish them anyway - especially as we got closer to the power supply voltage.
Not so in the digital realm - if the analog circuit doesn't do too much damage the digital part will faithfully reproduce what ever we feed it - or create. So we really need peak responding meters in our software, we just need to remember that they do not give us much of an idea of loudness. (A really short peak can reach the power supply rail and be effectively inaudible. How's that for trickery?)
Last thing for tonight (my finger are getting tired) - channels do not add algebraically unless the signals are identical land in phase. Say what?
Well if you've followed along this far you'd know that if I add two identical signals together I'll end up with a signal that is 6 dB louder. But individual tracks are not identical (who really wants to listen to 16 tracks of 1 kHz Sine wave anyway?)
So they don't add up, and in fact can be more - or less - than the 6 dB difference one might expect.
By the way, using sine waves is an old trick and very handy - if I apply a -6 dBu sine wave to two tracks and then send them to a buss the buss will measure 0 dBu. It's a terrific trick for balancing a stereo circuit, or two monitors. And we used to use tapes with a single track across two head gaps to align tape machines using the same principle.
I know that's a lot, and I know there is a lot more, but I thought this might be a good starting point.