/u/caffarelli already answered, but I guess I can add a couple things (focusing the instrumental side of "classical music").

This is a musical tradition. The term covers a hell of a lot of things. As caffarelli mentioned, art music is commonly used. Art music doesn't work for some of the things "classical music" covers , but for others it works just fine.

Now, a tradition is passed. If you look at how it is at one point in time and then wait for a while, you will notice changes (like it happens with languages). Many elements of "classical music" have survived for a lot of time, some others have appeared and kept going strong, some have died.

Classical music of the late 19th century was already very different from Beethoven, classical music from the late 20th was completely different and so on. There's classical music DNA in contemporary "popular music," even if orchestras, operas, piano recitals and so on ceased to exist today, aspects of classical music have been absorbed by other musical practices. What happened to Latin? Well, we have Spanish, French, Italian, Portuguese, Romanian...

Classical music REALLY separated itself from other forms of music, it started in the late 19th century (well, Beethoven was already up to something) and the big push came in the 20th. New classical music became kind of alien for the average people, and if you consider how new technologies created new possibilities, and how people's lives changed in the early 20th... It might look like classical music is done, and many people have announced the death of the opera, the orchestra, the ballet, and classical music as a whole.

As caffarelli said, classical music has been mostly the music of the rich and the people doing decently well. The whole thing revolves around their tastes and values, it's never been the music of the peasants or factory workers. However, at some points you find more people around it than at some others.

Let's talk about orchestras.

They were the rage in the second half of the 19th century and the early 20th in the big European capitals, and pretty much all the "civilized" cities (all the recently independent countries wanted to have proper orchestras, like the European ones). There started this ferocious competence in the US between between the end of WWI and the Great Depression, concert halls, record labels and radio stations were the battlefields for many orchestras: those from Cleveland, Detroit, and Los Angeles. They toured, made recordings... There were many new orchestras created by broadcasting services (BBC, Radio France, Deutsche Rundfunk, Radio France).

The big boom in the US started in the 1930s and lasted until about the 1980s. I think it's important to mention that the orchestral repertoire pretty much froze in time in the early 20th century. Modern music didn't really manage to get a foothold in the concert halls, it's mostly canon music (Mozart, Beethoven Brahms, Tchaikovsky, etc.).

Virgil Thomson offered a catalog of statistics on how Americans took their leisure, listing national expenditures of $45 million on orchestra music (paid “at the gate” in the form of 300 million admissions), baseball $40 million, horse and dog racing $38 million, professional football $9 million. Additionally $60 million was spent annually on classical records. Thomson counted 938 symphony orchestras in the United States. Music in general (teaching, performance, instrument manufacture) was the sixth largest American industry, after automobiles but before steel.

The boom was international in scope. For the most part the old European orchestras recovered within a few years from the rubble of World War II, so anxious were patrons and governments to resume their cultural programs as proof of returning normalcy. Orchestras based on European models had sprung up long before in colonial capitals from North Africa to Shanghai and Hong Kong; the 1950s saw healthy symphony orchestras emerge all across Asia, for instance the Central Philharmonic Orchestra of Beijing (1956) and the Seoul Philharmonic (1957). Tokyo alone, building on the success of the NHK (Radio) Orchestra (1926), boasted more than a half-dozen professional orchestras, with the halls, festivals, and music schools that have traditionally characterized the world’s great musical capitals. The thirst for orchestral music in the Asian markets of the 1970s and ’80s was all but unquenchable.

That was from Kern Holoman's Very Short Introduction to the Orchestra. It's a short book, it includes some details about the relationship between money and orchestras. you might find it very interesting.

What happened then? Well, shifts in the music industry as a whole, big economic crises... Orchestras were at some point paid for some rich folk, then several rich people, then a bunch of well doing people plus the fellows buying the tickets, then the government plus some rich people plus the ticket payers. Changes in taste, new media, the relevance of the whole ritual for the social life of the rich fellows. It doesn't help that it became a super star system, just like the movies. You will find Holoman's book very interesting.

Let's talk about pianos.

The piano had a very important place in the world of classical music. It was also part of the lives of the people who were doing well. It even became important for the working class who were suddenly able to have some nice things. See, well doing people had pianos, you had to keep up with the Joneses...

This started in the second half of the 19th century. There were a hell of a lot of piano makers in the US. People bought pianos even if they weren't able to play them.

What happened? Well, WWI (lack of materials), then the Depression (people struggled to buy pianos), then radio took over (why should I get a piano?) , then came recorded music (I can listen to whatever I want), WWII didn't make things easier... And we get to a completely different world.

In the late 20th and early 21th century we find less and less new pianos being sold, by a handful of makers (there were LOTS of brands in the late 19th and early 20th century).

Things might be changing. China has grown a lot since the late 20th century, and classical music became a symbol of status. If you are doing well, your child will learn to play an instrument (piano and violin are still up there in the food chain) and will play classical music. Because that is the music of well doing people, that is the respectable thing to do. China started making pianos in the 20th century, at first they were shitty. But they have been improving, and they started making a hell of a lot of them. They also need to make other instruments, and they want to have decent orchestras. It's not an accident that you see more and more good Chinese musicians.

Classical music lives a different life now. Things change.

For example, you can now find it in the cellphone of the geeky girl who goes to school in a not so well doing neighbourhood. She discovered it by accidenton Youtube, and got a hell of a lot of hours of music for her phone. That had NEVER happened in history.

Many countries have projects to use classical music for non-exclusively musical goals. Students come from poor socio-economic backgrounds and get into this kind of music, make friends, and work hard. They might not become professional musicians, but they keep away from trouble, learn to be responsible and to have a stable schedule to work on things, they develop cognitive skills. And you can be damn sure that they will learn something about classical music if they keep playing it for a while.

Still, not the music of most people, but it's there and it's getting to new places.

TL;DR

Classical music has never been really the music of most people. As time has passed, it has become more common among the population, but super popular might not be the the best description. Orchestras and piano were big, and then the modern world happened. This music is not quite dead, and if you look at numbers, they are bigger than ever. Even if all classical musicians died, there would still be classical music stuff surviving in many other forms of music.

The term you’re kinda looking for here is art music, though it’s a controversial term, so if you aren’t comfortable with using it, neither am I really, so I don’t blame you. The odd man out in this is jazz, which I would argue is consumed as art music now, but not necessarily during its “lifetime.” I’ll let /u/origamitiger answer for jazz. But opera is a subset of general art music, so they can go together.

Now I am going to tell you a dark secret. This dark secret, when it surfaces here, sometimes causes some ruffled feathers, disbelief, incredulity, and general pushback against it. This dark secret deeply goes against how opera and composers are presented in popular media. I hope your body is ready for this dark secret. Art music has never been very popular. It is a minority product for a minority people, almost entirely funded and consumed by the rich, or, at the very least, funded by the “oh well I’d say we’re comfortable.” It is at best only occasionally useful to compare consumption of music in the 18th and 19th centuries to modern music consumption now, they are so different, not to mention our class strata is so different too. Basically Mozart was not Elvis.

Here’s a less dark secret though. In terms of what’s best for consumers, you may be living in the best of all times to consume art music. Art music today is cheaper and more accessible than it ever was, with more variety, and more global accessibility to the best performers in the age of easier travel. I’m just an average working gal really, but a date night for me and my husband to see a top soprano in concert, which I’m actually planning right now, is not even the entirety of my day’s wages. Nowhere in history comes even close to this level of financial accessibility to art music. About 1-2% of Americans today attend the opera in a given year (source for 2012) (levels fluctuate with the economy and other things), which is really quite good historically. (This doesn’t even count the non-traditional ways of consuming it, in recordings, TV, live in movie theaters, etc., which the study I linked above put at 4%, which is crazy high compared to other historic rates of going to the opera) It was much less in Handel’s time. It might have gotten as high as 3-4% during the “reign” of the impresario Barbaja (which was the time of Rossini) but that would be a very rough estimate on my part, and that would only be during his really really successful times. The majority of the time for him was probably 1-2% like modern Americans, or even lower.

Reports of art music’s death have been greatly exaggerated. Just because you’re not the 1% consuming it doesn’t mean no one’s consuming it. There’s lots of great composers alive and working, and new works are still “entering the canon,” you’re just not likely to be exposed to them unless you’re active in the art music scene already. Likewise, I do not get Jandek but presumably someone is consuming it.

So… probably right now is the best time. Art music is currently full of excellent talent, it’s pretty well-funded, and it’s cheap to you. You’re living in a golden age! :)

Jazz really started to become popular in 1917, with the first recordings of the Original Dixieland Jazz Band (ODJB) becoming massively popular within a short time of their release. This white band had the effect of making jazz approachable to much of the middle class, white American society that had the disposable income to support the new music. Since 1910, James Reese Europe had led some important groups, playing a style of highly syncopated marches in New York. This group was one of the fore-runners of what is now called jazz. However, groups like Europe's, and others of a similar nature in New Orleans and Chicago, were confined to their respective cities. The marriage of the ODJB's new and exciting music with the quickly spreading power of recorded music thrust jazz on to the national stage, so to speak.

So, I'd say it started to become mainstream in 1917, and had accomplished it by 1920 or so. Now, the decline of jazz as popular music came about after WW2. By 1948, many of the formerly popular big bands had been forced to disband for lack of work, and Rock-and-Roll/R&B were spreading like wildfire.

I'm a big James Reese Europe fan, so I'mma take the time to mention that by 1918 he was leading a military band in France that had a very strong influence on the European music scene. He brought jazz-like music to France, and it quickly spread to the rest of Europe.