The deciBel (dB) is a method of expressing the ratio between two quantities in a logarithmic fashion. Used when describing audio signal amplitudes because the logarithmic nature matches the logarithmic character of the human sense of hearing.
The Decibel is used when comparing one signal level against another (such as the input and output levels of an amplifier or filter). When the two signal amplitudes are the same, the ratio is 1 and the decibel value is 0dB, from the formula dB = 20x Log (V1/V2). If one signal has twice the amplitude of the other the decibel value is +6dB, and if half the size it is -6dB.
When one signal is being compared to a standard reference level the dB term is supplemented with a suffix letter representing the specific reference in use. 0dBu implies a reference voltage of 0.775V rms, while 0dBV relates a reference voltage of 1.0V rms. The latter seems an obvious reference to use but, for complicated historical reasons, the 0.775Vrms value is used more often in professional audio.
In European broadcasting circles, the standard audio reference level is 0dBu (0.775Vrms), but American broadcasters chose a higher standard reference level of +4dBu (1.228Vrms) and used that level as the 'zero' for the VU meter. As American mixing consoles and outboard equipment grew in popularity in the UK and Europe, most music recording studios adopted the same standard, and that is now ubiquitous.
So the two most common standard audio level reference levels are +4dBu (1.228V rms) for balanced audio connections between professional equipment, and -10dBV (0.316V rms) for unbalanced connections on semi-pro equipment. Note that although the numerical difference between these values is 14, the actual signal level difference between these standards is 11.8dB (normally rounded out to 12dB), because the numbers use different reference signal voltages (dBu versus dBV)
The term dBm is also sometimes encountered on equipment specifications, but it is usually being used erroneously. The dBm relates to an amount of power rather than a signal voltage, specifically 0dBm implies a power of 1mW dissipated into 600 Ohms. This 0dBm reference was originally employed in the early telephone industry where it was necessary to transmit power from the telephone transmitter to a distant telephone receiver. As it happens, when 1mW of power is dissipated in 600 Ohms, the voltage across tat termination happens to be 0.775V rms — and that's what we still use as the reference for dBu, but without the need for a 600 Ohm load (the 'u' actually stands for 'unterminated').
When discussing acoustic sound levels, 0dB SPL (sound pressure level) is the typical threshold of human hearing at 1kHz, and relates to a sound pressure of 20 microPascals. 1 Pascal of sound pressure is equivalent to an SPL of 94dB, and 1 atmosphere of pressure is around 194dB SPL.