Decibel
Decibel is a unit of measurement that expresses the logarithmic ratio of two physical quantities of the same dimensions. The logarithm is to base 10. This logscale definition is useful when the quantities have a wide range and losses or gains are proportional.^{}
Decibel is dimensionless since it's a ratio.^{} It's a relative measure. However, when a reference is defined, it can be used as an absolute measure. As a relative measure, it's represented as dB. As an absolute measure, often an additional suffix is appended to "dB".^{}
Decibel originated in telephone networks of the early 20th century. Today it's commonly used in many domains including acoustics, electrical engineering, signal processing, RF power, and more.
Discussion

Could you explain the decibel scale with some numbers? Decibel is defined as \(10\ log_{10}(P_1/P_0)\), where power level P1 is compared against P0.^{} When P0 and P1 are equal, we say their ratio is 0dB. Thus, 0dB doesn't imply zero power or intensity. It implies equal power.
If P1 is twice the value of P0, we say that P1 is 3dB higher than P0. If P1 is half the value of P0, we say that P1 is 3dB lower than P0, or equivalently 3dB.^{}
For ratios P1/P0 equalling 10, 100, and 1000, the respective decibels are 10, 20, and 30. Likewise, if the ratios are 0.1, 0.01, and 0.001, the respective decibels are 10, 20, and 30. Thus, we can see that even though the power levels are changing geometrically, the equivalent decibel values change linearly. This is why the logarithmic scale of decibel is useful when quantifying values that have a large range.

What are some examples where the decibel scale is used? Decibel is used to quantify the intensity of sound. For sound, it's common to use decibel with a reference of 0.02 mPa (millipascals), the minimum sound that the human ear can hear.^{} Normal speech is at 60dB. A jet engine is at 120dB, which is a million times louder than normal speech.^{}
In electrical engineering and radio engineering, decibel is used. For example, the gain of an amplifier or the loss of signal power due to an obstruction are quantified in decibels. A directional antenna will focus its radiation in specific directions, which are specified on a chart with decibel as the unit.^{}
In signal processing, it's common to apply filters to signals. Bode plot is an example that shows the magnitude response of such a filter: decibels (yaxis) vs frequency (xaxis). This tells which frequencies are allowed to pass and which ones are attenuated.^{}
When a signal is compared to noise, called SignaltoNoise Ratio (SNR), this ratio is specified in decibels. For example, we can compare a processed image with the original and quantify the difference as SNR in decibels.^{}

What's the difference between decibels based on either rootpower level or power level? Decibel is commonly defined in terms of power levels. However, it can be defined in terms of signal or field level. For example, in electrical engineering, electrical signals are voltage and current. Power is proportional to square of these signals. When decibel is therefore defined in terms of voltage, the equation becomes \(20\ log_{10}(V_1/V_0)\), where voltage level V1 is compared against V0.^{} ^{}
While the term field quantity was previously used, ISO Standard 800001:2009 introduced the term rootpower quantity.^{}

What are absolute and relative decibel units? Decibel is a relative measure that compares two quantities of the same dimension. But in acoustics, it's common to say that noise was at 90dB. We don't say noise was 90dB with respect to something. This is because a reference of 0.02 mPa (millipascals) is implied since it's the quietest sound that we can hear. Thus, sound levels are in absolute decibels.^{}
In most other applications, dB is a relative measure. When we say that path loss of a wireless channel is 120dB, we mean that from the transmitter to receiver, signal power drops by 120dB. However, if we take 1 milliwatt as the reference, then we can use the unit dBm to indicate absolute measure. If transmitter sends a 10 kilowatt signal, it's sending a 70dBm signal. If path loss is 120dB (relative measure), the receiver will get a 50dBm (absolute measure) signal.^{}
Here are some examples of absolute measures (reference in parenthesis): dBW (1 watt), dBi (isotropic antenna), dBV (1 volt),^{} dBµV (1 microvolt), dBµV/m or dBu (1µV/m electrical field strength),^{} dBA ("A" weighted pressure levels),^{} and more.

What are some limitations of using decibels? In image processing, SignaltoNoise Ratio (SNR) and Peak SignaltoNoise Ratio (PSNR) are sometimes used to compare a processed image with the original image. SNR and PSNR are expressed in decibels.^{} The problem is that image quality is a matter of human perception. We may sometimes perceive one image as better than another even though it has a lower SNR.^{} Admittedly, this is not a limitation of decibel itself but rather a limitation of its usage for SNR.
There have been suggestions that decibel is outdated for the modern era. This is probably due to incomplete understanding of logarithms and incorrectly mixing absolute and relative values.^{}
Milestones
Scottish mathematician John Napier (aka Neper) publishes a book detailing his invention of the logarithms.^{} This would prove useful centuries later in the definition of the decibel.
In the early days of telephony, there's a need to quantify transmission losses due to links and nodes in a connection. AT&T proposes to use Mile of Standard Cable (MSC), which is equivalent of a mile of drycore cablepair with a loop resistance of 88 ohms and a mutual capacitance of 0.054 farads.^{} Let's note that MSC is not a measure of distance, but one of loss, useful for comparing the efficiencies of two telephone circuits.^{}
MSC unit has dependency on frequency. This was useful in earlier decades when there was a need to characterize distortion. With newer circuits having much less distortion that the standard circuit, there's a need for a distortionless unit.^{} ^{} Thus, AT&T invents the Transmission Unit (TU) to replace MSC.^{} TU is defined based on the logarithmic scale. The logscale is useful since the losses due to two successive parts can simply be added instead of multiplied.^{}
The International Advisory Committee on Long Distance Telephony in Europe, plus representatives of the Bell System, decide to adopt two units based on TU: bel based on power ratio \(10^1\) and neper based on power ratio \(e^2\). Since TU is based on power ratio \(10^.1\), the Bell System decides to use decibel as the unit.^{} One decibel is the smallest difference in sound that the human ear can detect. The word "bel" itself is honouring Alexander Graham Bell, inventor of the telephone.^{}
By this time, in the UK at least, the unit of loss measurement becomes decibel rather than msc, though some other countries continue to use neper as the unit.^{}
1937
At the First International Acoustical Conference in Paris, decibel is adopted as an international unit for energy and pressure levels.^{}
2003
There's talk of including the decibel within the International System of Units (SI) but this proposal is rejected. However, decibel has been recognized by IEC and ISO.^{}
References
 Admiralty. 1938. "Appendix A: The DeciBel and the Neper." Admiralty Handbook of Wireless Telegraphy, B.R.230 (Vol. II). HMSO London. Accessed 20190605.
 All About Circuits. 2019. "The RF Engineer’s Guide to the Decibel." All About Circuits. Accessed 20190605.
 Giangrandi, Iacopo. 2000. "Power and amplitude: Watts, Volts and referenced Decibels." September. Updated 20170501. Accessed 20190605.
 Gregersen, Erik. 2019. "Decibel: Unit of Measurement." Encyclopædia Britannica. Accessed 20190605.
 Hall, David A. 2014. "All About Decibels." Electronic Design, January 16. Accessed 20190605.
 Hartley, R. V. L. 1928. "TU Becomes Decibel." Bell Laboratories Record, vol. 7, no. 4, pp. 137139, December. Accessed 20190605.
 Herbozo, Gery. 2014. "Decibel = 'old era'? If so, what is the unit for modern digital processing?" ResearchGate, March 03. Accessed 20190608.
 HistoryComputer. 2019. "The Logarithms and Rules." Accessed 20190605.
 HowStuffWorks. 2000. "What is a decibel, and how is it measured?" HowStuffWorks, April 01. Accessed 20190605.
 IAC Acoustics. 2019. "Noise Control FAQs." IAC Acoustics. Accessed 20190605.
 Lin, Weisi, and C.C. Jay Kuo. 2011. "Perceptual visual quality metrics: A survey." J. Vis. Communi., Elsevier, vol. 22, no. 4, pp. 297312, May. Accessed 20190608.
 Martin, W. H. 1924. "The Transmission Unit and Telephone Transmission Reference Systems." Bell System Technical Journal, vol. 3, no. 3, pp. 400408, July. Accessed 20190605.
 Martin, W. H. 1929. "Decibel  The Name for the Transmission Unit." Bell System Technical Journal, vol. 8, no. 1, pp. 12, January. Accessed 20190605.
 RadioTimetraveller. 2015. "The dBµ vs. dBu Mystery: Signal Strength vs. Field Strength?" Blog, RadioTimetraveller, February 24. Accessed 20190605.
 Sage, Daniel. 2017. "SNR, PSNR, RMSE, MAE." Biomedical Imaging Group, EPFL, July 30. Accessed 20190605.
 Sizes. 2014a. "decibel." Sizes, December 09. Accessed 20190605.
 Sizes. 2014b. "mile of standard cable." Sizes, May 30. Accessed 20190605.
 Ward, Keith. 2006. "A Short History of Telecommunications Transmission in the UK." The Journal of The Communications Network, vol. 5, part 1, pp. 3041, January–March. Accessed 20190605.
 Wikipedia. 2019. "Decibel." Wikipedia, May 17. Accessed 20190605.
 Wolfe, Joe. 2019. "dB: What is a decibel?" Physclips, School of Physics, UNSW, Australia. Accessed 20190605.
Further Reading
 Martin, W. H. 1924. "The Transmission Unit and Telephone Transmission Reference Systems." Bell System Technical Journal, vol. 3, no. 3, pp. 400408, July. Accessed 20190605.
 Sizes. 2014a. "decibel." Sizes, December 09. Accessed 20190605.
 Hall, David A. 2014. "All About Decibels." Electronic Design, January 16. Accessed 20190605.