The word "classical" as applied to music has two meanings. The narrow meaning is the music that predominated in cultivated circles from about 1759 or so (the year Handel died) through about 1827 (the year Beethoven died). The most prominent classical composers were Haydn, Mozart, and Beethoven, though there were many others as well--among them J.S. Bach's sons, esp. C.P.E. Bach, Johann Nepomuk Hummel, Luigi Boccherini, Karl Ditters von Dittersdorf.and many others. Of course, this time classification is a bit arbitrary; Schubert, who died a scant 20 months after Beethoven, is generally considered a Romantic composer, though many of his longer works from all but the last few years of his life are basically classical in the narrow sense.
In the broader sense, classical music is more difficult to define precisely; you can find exceptions to almost any generalizations you make.
Yes, classical music tends to be in longer forms that popular music, but a great many classical composers have written songs which are no longer that the average pop song. There is even an Australian composer named Myroslav Gutej whose 3 movenment Piano Sonata #7 lasts a total of 24 seconds!
Yes, classical music tends to be "serious" music; in fact, many have adopted the term "serious music" in lieu of classical music, because they think it more accurate. But that does not mean that all non-classical music is not serious, or that no classical music has humor. Many operas have comic plots; probably about half of opera plots involve crossdressing by at least one character. And Brahms Academic Festival Overture, written, as you might expect, to celebrate a particular academic festival, horrified many of the academics and delighted their students because it was based on four popular German student drinking songs. Much of the humor of J.S. Bach is lost to us now, because much of his music, including his sacred cantatas and organ music, incorporated melodies from the popular music of his day; this often shocked many members of the congregation. At the other end, the song "Strange Fruit," popularized by Billie Holiday, certainly qualifies as "serious" music.
And finally, whether a composer is considered classical or popular can be a rather arbirary, individual decision at the margins. I have made the decision, for example, to include all of Duke Ellington, even his longer symphonic compositions, in with my jazz section. OTOH, I categorize both George Gershwin and Scott Joplin (not just his opera Treemonisha) as classical composers.
And while I think rap is an abomination, I must say that many people who began as rap "artists" have turned out to be people of real artistic depth in music and movies. Queen Latifah is one of the few who has branched out into other kinds of music, but others, like Ice T and Will Smith have shown themselves to be fine actors with genuine depth. And, divorced from its often obscene, violent, and mysogynistic content, rap as a form, if not a name, has been around for a long time, going back at least as far as the Gilbert and Sullivan patter songs. And give another listen to Pete Seeger's "Talking Atom" from the late 40's, if you have never heard it.
Favorite essentials to a good classical CD collection:
Bach: Brandenburg Concerti--Marriner, cond.
Handel: Messiah--Charles Mackerras, cond.
Haydn: String Quartets, Op. 76--Tatrai Quartet.
Beethoven: Piano Concerti (5)--Leon Fleisher, piano, George Szell, cond. Cleveland Orchestra.
Brahms: Symphonies--Georg Solti, Chicago Sym.
Mahler: Symphonies--Bernstein, NYPO
Dvorak: Sym 9 "New World"--Zdenek Macal, cond, London Philharmonic Orch.
Stravinsky: The Rite of Spring--Stravinsky, cond.
Joplin: Treemonisha--Gunther Schuller, cond.
Prokofiev: Symphonies 1, 5--James Levine, Chicago Symphony.
Shostakovich: Piano Quintet--St Petersburg String Quartet, hyperion label.
Britten: Young Person's Guide to the Orchestra--Antal Dorati, cond., Sean Connery, narrator.