In today's world one can listen to any piece of music from anywhere pretty much at the click of a button and we have radio and other methods to hear music we are not familiar with. However this was obviously not always the case and before recordings music always had to be preformed.
So how did music propagate through society then especially in the context of music that was deemed "popular"?
Oh, I see. Here's that story:
During the time before music printing, and well into it, the vast majority of people didn't know about any music that wasn't being made right in their town or village. And there were very important composers (J.S. Bach is maybe the best example) who were completely obscure outside the city or small court in which they worked. For most of the period from 1000-1700, the Catholic or Protestant church was the main way music was propagated across Europe. Individual church composers (like Josquin or Palestrina) moved around inside the Catholic world, trying to get the best gig, and took their music with them. Meanwhile, Protestant church authorities were interested in publishing new hymns as a kind of evangelism for the new churches they were forming. (The first music book printed in the US was a book of Protestant hymns.)
BUT...
A composer like Beethoven had to move to the capital city of the Austrian Empire because that's where a large population of wealthy, cultured, aristocrats lived who would go to his concerts, have their children study with him, and (later) commission works from him and buy his printed music. Some musicians, like Mozart when a child virtuoso, toured like today's rock bands -- Mozart and his family made a number of European tours, which were very complex to arrange, needing the patronage of connected aristocrats to provide entrée to royal courts, etc. Their reputations spread along with them, and for some of them, publishers of sheet music found it worthwhile to print up editions of their more popular compositions, especially piano, vocal, and chamber pieces that amateurs could play.
Then these publishers, er, publicized their catalogs, offering the music for sale across Europe through importers and distributors. Some of them subsidized magazines for musical cognoscenti, in which reviews of new works and articles about musical life were published. A number of famous composers (Schumann, Berlioz) wrote for these magazines, and there are several famous articles where, in effect, an established composer says, "hey, check out this new guy!" (Schumann did this for Chopin in Germany (he was a Parisian virtuoso) and for a very young Brahms.) These magazines circulated pretty widely among the middle and upper classes, so, if you were interested, you could keep up with new music that way.
Publishers also collected "favorite pieces" into albums and sold those. A young girl studying piano in the time of the Jane Austen novels might have bought for her a large bound volume filled with piano solos, easier opera songs, folk songs, and the like, so that she could entertain after dinner from the parlor piano. Some composers got big exposure that way.
Finally, opera was a huge business in Europe all during the 17th, 18th, and 19th centuries: composers of opera like Rossini became genuinely and internationally famous. They wrote their operas to be premiered at and the exclusive property of one opera house for a given season, but if the opera was a success there, other houses across Europe would put it on for their patrons. Rossini was, at the height of his fame in the 1820s and 1830s, famous "from London to St. Petersburg," and his operas were put on in most major and regional capital cities. Popular arias from his operas were printed and sold as sheet music, and made it into those collections I mentioned earlier.
By the late 1800s, the combination of movable music type and steam-driven printing presses made sheet music -- for the first time -- cheap enough to be profitable as a mass market item. This was also the period when every middle-class home had some kind of piano in it. By the early 1900s, the music business starts to look like any other business, with manufacturers making large runs of product, distributors pushing the product out to regional markets, and dedicated "music stores" where consumers could browse the shelves for new music to buy. Before there were recordings (and even after), "song pluggers" were hired to go around and play new songs for customers at shops, at theaters and vaudeville houses, even at the new "moving picture" shows, where they sometimes used magic lantern slides to project the words and have sing-a-longs with the audience.
And then came the record business...
As well as the use of printing and notation (As described in other answers), there has also always been an oral/aural tradition of one performer simply passing on a song or tune to another. This would have been the predominant way that folk music, as opposed to art music or sacred music, was passed about. This is why a lot of folk tunes have a number of different versions relating to different places the tune had passed though (an old fashioned version of remixing, maybe?). Performers themselves would also travel far and wide, in the tradition of the "wandering minstrel".
As for passing on knowledge about art music - while it's true that there wasn't the radio or youtube, families with the money to purchase instruments would do so as they were one of the relatively fewer ways you could have some fun at home. With those instruments would come lessons which would pass around the knowledge of music repertoire, which would be then become known in that level of society. Likewise, if you were entertaining, chamber music was a form of entertainment designed to work in a room in your house. If you were to go out to see some entertainment, it would often involve some form of musical performance - perhaps a concert or operetta.
Even if you just went down to the pub, by the early 19th century, there may well have been some kind of musical performance going on - these performances would go on to evolve into the music hall and vaudeville traditions.
On a more sinister level, sometimes whole populations would move - or be moved - for reasons beyond their control. Hence the African influences we see in American music today.