Taken from a previous answer:

Firstly, one thing to remember about 'popular music since the 1950s' and 'modern bands' is that popular music generally reflects the times, in various ways, because it thrives on what is perceived as novelty. And Charlie Gillett makes the argument quite strongly in the book The Sound Of The City that rock'n'roll is, well, the sound of the city - a place full of mechanisation, loud vehicles, a place of bustling huge amounts of people. In the 1950s, too, the city had long been a place of electricity.

And so, unsurprisingly, one defining feature of all of rock instrumentation was that the instruments were just loud. Some instruments were amplified using specialised amplifiers powered by electricity - see electric guitars, electric bass, combo organs and electric pianos. And some rock instrumentation is just really loud - see the modern drum kit, or the modern concert grand piano, especially when played with the very percussive piano playing style used by Little Richard and Jerry Lee Lewis. Additionally, microphone technology had improved enough by the 1950s that live singers could actually be heard over all the noise of the drums and guitars.

In general, the standard instrumentation of rock'n'roll was all pretty new in the 1950s. Drums have obviously been around since time immemorial, but the idea that one person might sit at a kit of drums and play them in combination was relatively new, derived originally from the need for sound effects to accompany silent films played in theatres. These early drum kits were eventually crafted into something that looks like a modern drum kit in the 1930s and 1940s by jazz drummers like Gene Krupa (rock'n'roll drummers very commonly had a background in jazz; Charlie Watts of the Rolling Stones famously prefers jazz to rock'n'roll).

Electric guitars also became popular in the 1930s and 1940s. There were often acoustic guitars as a part of swing bands, but the relatively quiet volume of an acoustic guitar (in a largely unamplified band revolving around a large horn section) meant that the acoustic guitar was largely a rhythm instrument. However, an electrified guitar allowed jazz guitarists like Charlie Christian towards the end of the 1930s to actually play solos on the guitar which could be heard by the audience. Electric guitars thus spread through genres like jazz and rhythm & blues in the 1940s and 1950s because of their versatility - you could use them to play rhythm parts or lead parts (or in the case of Jimi Hendrix in the 1960s, both at the same time).

The electric bass guitar (i.e., the one that looks like a slightly bigger electric guitar, held like a guitar), as opposed to the acoustic upright bass (the one that looks like a big violin that's often taller than the person playing it, held upright), first went into production in 1951 (the Fender precision bass). Bill Black in Elvis Presley's band, for example, switched to electric bass in 1957.

Finally, electrified keyboard instruments gained prominence in the 1950s as well. The Fender Rhodes electric piano and Wurlitzer electric piano functioned on similar principles to an electric guitar, except with hammers hitting metal tines rather than plectrums hitting metal strings. The Wurlitzer went into production in the mid-1950s, and Leo Fender of Fender Guitars partnered with Harold Rhodes to mass-produce Fender Rhodes keyboards from 1959. Ray Charles prominently used the Wurlitzer electric piano on his 1957 hit 'What'd I Say'. On early Beatles recordings, a Hohner Pianet - a similar instrument made by the German company Hohner - was often played by George Martin to subtly supplement the rest of the band. I discuss the use of the electric organ in 1960s pop music in much more detail here but in general, electric organs are also similar to electric pianos and electric bass guitar in that they came to prominence in the late 1950s.

This line-up of instruments developed in rock'n'roll in the 1950s, often in bands attempting to emulate swing line-ups of various sorts. Bill Haley & The Comets had originally been a western swing band, while Ike Turner & The Kings Of Rhythm was an offshoot of a more straight-ahead swing big band; these musicians were attempting to emulate a big band with a smaller line-up using louder instruments. This line-up became codified as the standard in the 1960s, in particular because of the sheer success of groups like the Beatles, the Beach Boys, the Rolling Stones, and Bob Dylan. All of these line-ups basically used line-ups comprised of these instruments, perhaps supplemented by others (e.g., the Beatles using a symphony orchestra on 'The Day In The Life', the Rolling Stones using a children's choir on 'You Can't Always Get What You Want', and The Beach Boys using bass harmonicas and French horns amongst other things on Pet Sounds).

Guitars and electric keyboards are both versatile instruments with a reasonably large range of available pitches, where one musician is capable of both rhythm and lead parts which can both be heard by the audience. In comparison, a single saxophone (also a very common instrument in popular music since the 1950s) only allows for a single melody line rather than chords doing rhythm parts. Swing bands do show the full capability of saxophones and trumpets to do rhythm parts, of course, but this requires a relatively large horn section. It's not coincidental that it's only when polyphonic synthesisers and programmable drum machines become relatively versatile, affordable, and compact (and as computers rise in prominence in society in various ways in general) in the late 1970s and early 1980s, that synthesisers and drum machines really become dominant part of popular music.

Of course, much modern popular music very often doesn't include electric guitars, electric bass, acoustic drums, or acoustic/electric keyboards at all! It's often largely made in digital audio workstation programs on computers (e.g., Ableton, Logic, or ProTools) using digital samples and keyboards that - like the computer keyboards I typed this on - essentially input data to be processed by computer algorithms. However, such music also very often uses the logic of the modern band instrumentation, with a rhythm part usually conceptualised in similar ways to a drummer playing a drum kit, a bass part that's a single melody line, instruments playing chords in a rhythmic way, and lead lines (and vocals). But then this logic isn't dramatically different to the logic of the swing band, either - it just requires less people to carry it out. So where Duke Ellington's big band used fourteen members, a modern electronic act can just be one person.