This is obviously a big question covering centuries once you get into the issues around copyright and music scores; I'm going to specifically cover bootleg recordings, with a focus on rock music. Essentially, for bootleg recordings to circulate, you need people with portable audio equipment (or the ability to record radio broadcasts, etc), you need people able to convert those recordings into physical artifacts capable of being sold or traded, and you need a network whereby buyers can enter the market. And you need all of the above to be - if not legal, then either impossible to police, or where the police weren't particularly focused on policing it.

In this jazz world, by the late 1940s-early 1950s era of the saxophonist Charlie Parker, much of this ecosystem was in place in some ways, in that jazz aficionados were obsessed with Parker's solos, and would trade (somewhat primitive) recordings of Parker's live gigs - less with the object of making a saleable recording and more with the aim of recording the solo so as to analyse and then emulate it. After Parker's death in 1955, with these things circulating unofficially, the Parker estate put together a record company, Charlie Parker Records, to officially release these amateur recordings for fans, which released its first albums in 1961 (e.g., Bird Is Free).

In the context of popular music, by the early 1960s, home audio recording equipment was becoming cheaper and higher in quality in Anglophone countries, and recordings begin to circulate. Some of the recordings on the Beatles' 1994 Live At The BBC release, which featured recordings of sessions the Beatles recorded for the BBC to be played on radio, did not exist in the BBC's archives. As a result, the recordings released in 1994 in some cases came from private collections, where Beatles fans had recorded the recordings on their home equipment - the earliest recording of the Beatles from the BBC circulating in trading circles is from a March 1962 performance on the program Teenager's Turn, where they do versions of 'Dream Baby', 'Memphis Tennessee' and 'Please Mr Postman' - six months before they released 'Love Me Do', with Pete Best (audibly) still the drummer.

Similarly, a Beatles performance (or several performances, more likely) at the Star Club in Hamburg, from 1962, were recorded with audio equipment in the crowd, by a member of another British band playing in Hamburg at the time, and because the Beatles weren't under contract at the time, and they gave consent for the recording, apparently, the Star Club material was released semi-officially in the late 1970s, despite its amateurish recording quality, to be regarded as a historical document more than something listenable.

The bootleg wasn't really considered much of a commercial proposition in the rock world, however, until the release of the Great White Wonder vinyl bootleg in 1969. The context of this bootleg was that, in 1967-1968, Bob Dylan - one of the biggest rock stars in the world at this point - had recorded a set of new songs (now widely called the Basement Tapes) in a basement with his backing band, The Hawks, and distributed the basement recordings on acetate pressings with the aim of getting other acts to record the songs. This was successful for Dylan; The Byrds recorded 'Nothing Was Delivered' and 'You Ain't Going Nowhere', Brian Auger, Julie Driscoll and Trinity recorded 'This Wheel's On Fire' (a version perhaps best known these days as the theme from Absolutely Fabulous), and Manfred Mann had a big hit with 'The Mighty Quinn', etc. Dylan's backing band The Hawks changed their name to The Band and in 1968 released an album, Music From Big Pink, which featured professionally recorded covers of other songs from the basement recordings (opening and ending the album with the songs Tears Of Rage and I Shall Be Released, the first release of what would become a folk standard).

At some point in 1969, 'Ken and Dub', two people with access to informal recording trading networks, pressed and released a 2LP vinyl bootleg called the Great White Wonder, which featured various Dylan recordings taken from a variety of sources. About half of the GWW bootleg features recordings of a 1961 live performance Bob Dylan did in someone's apartment, while there are a few studio outtakes on side two. But Side Two and Side Four prominently feature recordings taken from the publisher acetates of the basement recordings that Dylan had been shopping around to acts, and this caused a sensation; Jann Wenner in the editorial of an edition of Rolling Stone devoted the editorial to demanding that the basement recordings be released as an album, while cuts from the bootleg were played on radio stations. Eric Harvey, a Pitchfork writer and academic, has a good article on the Great White Wonder here, while there are some contemporaneous Rolling Stone articles about the bootleg here, here and here. Perhaps it was the time for the bootleg; after all, the hippies were quite happy to simply climb over the fence at festivals like Woodstock, rather than pay - music should be free, maaan.

All of this made the bootleg much more prominent than it had been, and indeed Dylan and the Band duly released an album in 1975, The Basement Tapes, full of (lightly overdubbed in some cases) basement recordings from 1967-1968, to rapturous critical acclaim. As a result of the acclaim, a network of record stores willing to carry bootleg records under the counter developed. By 1977, for example, the Sex Pistols - frustrated with the major label process and the lengthy recording sessions - semi-officially released Spunk, a bootleg of what are essentially early studio rough mixes of the music that would eventually be released on Never Mind The Bollocks Here's The Sex Pistols, complete with songs with altered names (e.g., 'God Save The Queen' on Spunk is labelled as 'No Future') through this bootleg network.

Additionally, the advent of the compact cassette, first released in 1962, but becoming much more popular in the 1970s and then 1980s, then revolutionised access to music, as the compact cassette significantly democratised 'home taping', as compared to the previous 'reel-to-reel' tapes, which had to be threaded carefully by users, which were much less durable than the tape stored within plastic cases in a compact cassette, and which was a fair bit more expensive.

By the 1980s, the sheer popularity of compact cassette was concerning the music industry, because you could buy cassettes cheaply and simply tape things off the radio, or you could very easily dub a copy of a record or tape depending on whether you had a double tape deck or a hi-fi system where you could simply feed a signal from a record player (or a compact disc player, as the 1980s progressed) to a tape deck. The sheer ease of this process and the relatively small capital outlay involved in making a bunch of tapes and photocopying some liner notes meant that it was relatively common for there to be a bloke at the local markets selling copies of tapes; to get anecdotal for a second, my childhood copy of Madonna's The Immaculate Collection, officially released in 1990, was one of these.

The music industry didn't look kindly upon all this anarchistic home taping, as you can imagine; the rather famous (and widely parodied) 'you wouldn't download a car' campaign of the 2000s has an ancestor in the (also widely parodied) 1981 British record industry campaign about how 'Home Taping is Killing Music - And It's Illegal, here profiled in a 1981 issue of Billboard'.

By the late 1990s, with the rise of commercially available (and eventually cheapish) CD recorders, and the mp3 format, it became possible to essentially perfectly replicate digital recordings and to distribute them in a relatively high quality form on the internet (in the case of the mp3), or via the internet (in the case of CD-R bootlegs posted by fans across the world). The rise of internet communication media like newsgroups, IRC, and web forums meant that fans of particular bands willing to go on the internet had relatively easy access to a network of people willing to trade live bootlegs or rarities. Similarly, a couple of years before the rise of Napster, on the internet, a network of FTPs and open directories on the net developed where people could download mp3s of albums. 1998 is the last year before the 20-year-rule, and if I might get anecdotal again, by 1998 I was personally downloading mp3s from newsgroups, IRC file transfers and FTPs; for example, here is an MTV article from 1998 about a Michigan Technological University student getting a cease and desist for hosting an archive of Smashing Pumpkins bootlegs and rare tracks; in the spirit of full disclosure, I have to say I spent a fair amount of hours in 1997 and 1998 patiently waiting for a 2MB 128kbps mp3 file of some Smashing Pumpkins rarities to download over my 28.8k modem.

I recommend Clinton Heylin's 1994 Bootleg: The Secret History Of The Other Recording History for more on this topic.