I said in a previous post that the term 'classical' music is a back-formation, meaning that until popular music became an economically huge force, there was just music with different flavors for different purposes. For listening, you went to the symphony or a chamber or lieder recital, for dancing you had Viennese walzes--just to choose a couple of examples. But once recording technology and television came along, popular music grew to dwarf what we now call 'classical' music. My view is that any music that has long-lasting value, of whatever genre or style, will sooner or later become 'classical' in the sense of "the best that has been done". As you may have noticed, I place the oeuvre of the Beatles in that category.
That aside, people new to classical music notice that it comes in different flavors. The repertoire of symphony orchestras roughly spans the period from when the symphony was invented, mostly by Joseph Haydn around 1770, to the present day, though in the current environment most composers do not choose the symphony as the main focus of their work. But we can divide this repertoire into phases. The Classical phase includes Haydn, Mozart and Beethoven--and possibly Schubert. But between 1820 and 1830 a new spirit came to be: that of Romanticism. This is a bit confusing because we use the word 'classical' in two senses. In one sense it is all that music that is not popular. But in the narrower sense it means music of a particular historical period: from 1770 to the death of Beethoven in 1827. The Romantic phase of 'classical' music starts from then and goes to the late 19th and early 20th centuries. Composers like Berlioz, Schumann, Chopin, Bruckner, Mahler, Liszt and Wagner are all Romantic composers.
Another interesting thing happened in the 19th century: there began to be more interest in the music of the past than ever before. This really started in 1829 when Mendelssohn organized a performance of Bach's St. Matthew Passion in Berlin one hundred years after it was first performed in Leipzig. Even though composers like Beethoven had studied the works of Bach, he was not well-known among the general listening public. This changed radically during the 19th and 20th centuries. Bach is from the period known as the Baroque, or what we might call pre-Classical classical music. It was not just Bach that was re-discovered. He brought along with him a host of other Baroque composers like Rameau, a couple of Couperins, Froberger, Corelli and, that bane of background music, the Vivaldi Four Seasons. After that, it was easy to develop an interest in pre-Baroque music and today we have lutenists and madrigal groups that go around playing Renaissance music. There are also ensembles that go further back to the Middle Ages. The wonderful thing is that we have discovered outstanding pieces of music going right back to the beginnings of Western music around the year 1000.
I won't talk about the discovery of non-Western music here, but that has been a trend for the last fifty years or so. But I should mention what came after the Romantic period. In the early 20th century there was a great paroxysm in Western Civilization. One symptom was World War I, followed by the sequel, World War II. Some have called this a "suicide attempt by Western Civilization". There were similar events in music as well. The organization of music in terms of the modes or scales was overthrown with the development of atonal music and the incorporation of percussion, noises and chance procedures. The names to note are Schoenberg, Berg and Webern on the one hand and Cage, Stockhausen and Varese on the other. The first group followed the traditions of using ordinary instruments but in an atonal context; the latter group incorporated new instruments, electronics, noise and chance procedures. It is still an open question whether much of this music will become 'classical' in my sense of the word. I'm sure that some will, but much won't.
So there you have a quick march through several different flavors of classical music. Now for some listening. Here is a piece from the early 20th century that I am fairly sure will become a 'classic' if it isn't already. The Lyric Suite for string quartet by Alban Berg, completed in 1926: