The decibel (dB) is a measure of the ratio between two quantities, and is used in a wide variety of measurements in acoustics, physics and electronics. While originally only used for power and intensity ratios, it has come to be used more generally in engineering. In electronics, we can use dB to describe the gain of an op-amp for instance.
The use of decibels has a number of merits:
It is more convenient to add the decibel values of, for instance, two consecutive amplifiers rather than to multiply their amplification factors.
Decibels are useful because they allow even very large or small ratios to be represented with a conveniently small number (similar to scientific notation). This is achieved by using a logarithm, allowing one to clearly visualize huge changes of some quantity. The formula below shows that dB can be used to describe op-amp gain (A).
For example, if the open-loop gain of an op-amp is 100dB, we have 100dB = 20 (log Vout/Vin)
100/20 = log (Vout/Vin)
5 = log (Vout/Vin)
(Vout/Vin) = 100,000
If the input changes by 1V, the output changes by 100,000V. This demonstrates the high open-loop gain of op-amp. Realistically, the op-amp output goes to the rail if the inputs change by 1V. Assuming the rail is 5V, it only takes the input voltage change by 5/100,000=50uV for the op-amp to go to the 5V rail.