Since the early 1900s, certain classes of instruments, especially plucked strings like guitars and basses, found themselves outclassed in volume by other common instruments of the day, such as brass sections and drum-kits of "big band" groups. Microphones were originally used to increase their volume, however they had several drawbacks including requiring the performer to keep the instrument in one place, and also picking up the sounds of surrounding instruments, further unbalancing the group. The answer eventually came in the form of the magnetic transducer, borrowed from the microphone and installed directly onto the instrument itself to sense the vibrations of the metal strings, sending that signal to the amplification system to be reinforced. These "pickups" would eventually be custom-designed for the instrument, and new instruments liek the solid-body guitar and electric bass guitar would be designed specifically to make use of the new technology while overcoming inherent problems of mounting transducers into acoustic instruments, namely feedback.
As these instruments gained popularity, the music industry was faced with other problems relating to the size and weight of traditional instruments. Pianos and other keyboard instruments were a particular concern; the sheer size and weight of a full concert grand, as well as its sensitivity to climate and simply to being moved around, made it impractical for touring acts to bring a quality piano with them, making the pianist of a jazz combo dependent on whatever instrument happened to be available at the venue. The bleedover of the console pipe organ from gospel music into rhythm and blues and jazz styles gave performers yet another expensive, delicate and temperamental instrument to have to transport or provide. The answer came with the electronic synthesizer, originally a large, clunky apparatus producing a very synthetic-sounding series of tones, but eventually evolving into relatively accurate simulations of organ pipes and eventually plucked and hammered strings. The earliest examples date back to the 1930s but wouldn't become common in mainstream music until the late '50s as a more portable substitute for acoustic keyboard instruments.
During this same time, audio amplification also evolved from its infancy in studio and broadcast uses into a mainstay of live sound productions. For much of the 20th century, amplification was achieved through the use of multiple independent stage amplifiers for each source (including vocals) that needed reinforcement, which were then captured by a single "ambient" microphone in front of the entire group The "mixer", a single console allowing a sound engineer to mix and blend various sources, was introduced first in the studio as a necessary consequence of the development of the multitrack recorder in the late 1960s, to combine these sources in a "mixdown" to a single blended signal. The idea would soon be applied to the multiple independently-amplified sources of a music group during a live performance, beginning in the 1970s.
The heart and soul of virtually every electronic audio circuit is the operational amplifier or op-amp, a transistor-based circuit which uses an input signal produced by a microphone or other transducer to control the flow of a larger secondary power source. These were first developed based on vacuum tubes, which evolved into "solid-state" transistors and then into integrated circuits "printed" on semiconducting materials. These op-amps, also referred to as "gain stages", are used virtually anywhere that the voltage level of all or part of an audio signal is modified by an audio circuit. Op-amps are incorporated into circuits alongside capacitors and inductors, which when given an alternating current like that of an audio signal, behave in "non-ideal" but beneficial ways, such as presenting higher resistance to different frequencies, and shifting the phase of the voltage of an output signal relative to that of the input. These allow audio circuits to isolate certain frequency ranges for amplification or attenuation, or to induce beneficial changes in phase that can be used to cancel out unwanted signals, and form the core of virtually all analog tone-shaping effects.
In more modern times, the detrimental side effects of analog circuitry are avoided by converting the signal into a digital format. A series of "samples" of the amplitude of the analog waveform are taken, typically tens of thousands per second, and these samples are converted into a number, which can be represented by a series of ones and zeroes, in turn represented by voltage variances in the line carrying the digital signal. When represented this way, artifacts that are typically added to an analog signal in the transmission line, such as electromagnetic noise, are avoided, because the noise doesn't change the values of the digits being sent along the line as it would an analog waveform. The digital samples can also be very precisely modified to simulate analog tone-shaping circuitry, without undesirable side effects such as attenuation, phase-shifting or "noisy" transformation caused by tolerance variances in components. The flip side of digital signal processing is that it can introduce its own artifacts, as the continuous analog signal is deconstructed into a finite, discontinuous series of samples, which are then used to reconstruct an analog signal for amplification. This digital process also requires that all the various pieces of equipment that transport and consume the digital signals do so in a synchronized fashion, otherwise a piece of equipment could misinterpret the digital signal and produce the wrong waveform values. Digital is also all-or-nothing; while an analog signal can become noisy, the sound is typically still intelligible at this reduced quality. If a digital signal becomes noisy or faint enough that the digits cannot be interpreted correctly on the receiving end, the signal becomes completely unusable.