Claude Debussy - Claire De Lune. If you type that in anywhere you'll get a lot of other similar Debussy pieces for piano, such as his many preludes and nocturnes, and they're all very beautiful.
Maurice Ravel - Bolero, J'eux D'eux
Igor Stravinsky - Rite of Spring, Firebird. Stravinsky wrote a lot of very interesting works for orchestra, all of which are worth hearing.
Philip Glass - 'Glassworks' collection, particularly Purite Igoe (no idea if I spelt that right). Also Music in 5ths.
Steve Reich - Some of his works are hard to appreciate, he's very experimental, but his more accessible works include 'Four Organs', 'New York Counterpoint', 'Electric Counterpoint' and 'City Life - Check It Out'. Some of his works don't sound like classical music which is something I like about him a great deal, very very forward thinking composer.
Terry Riley - In C, the best thing about this piece is you could watch a hundred performances of the same piece and no two will be the same.
Tchaikovsky - Romeo and Juliet is a piece of his that's very special to me, but I've not given the rest a listen lately.
Handel - Water Music is beautiful, and Music for Fireworks is very nice. Very famous pieces, find 'em anywhere.
Bach - One of the Classical 101s, everyone likes Bach, Beethoven and Mozart. Bach's got a huge resource of music, so can be difficult to get into. While I can't remember the name of the piece, he composed a piece of music for four harpsichords, which he had his students - who were not very talented - play, with a large focus on ornamentation and improvisation. Its a big big sound and its fantastic. I'll have a hunt for the cassette if you want me to, but I think its Harpsichord Concerto 17.
I have intentionally neglected to include the eras and movements of the composers I mentioned because some of them who you may not have heard of have this dreadful prejudice attached to them within the classical community, and I didn't know anything about it until I was already a big fan and I'm glad I had the opportunity to do that. If you know anything about classical music, or if you like these compositions, you'll probably find out regardless, but I'd like you to hear it free from preconception.
Uh, there's also a Japanese orchestra I like, who uh... Well, before every performance - all female orchestra - they get up, one by one, strip off all clothing and then start playing. Hehehe. They actually have a very interesting way of approaching it, they have a lot of electric instruments. But uh... I don't know if my affections are being SWAYED by anything in particular ;)
Cannibal Corpse - Hammer Smashed Face
Slipknot - Vermillion
Korn - Twisted Transistor
Static-X - I'm with Stupid
Mudavayne - Dig
I think this one's a bit too recent to be considered 'classical', but I think you might like it:
I love the Overture from Carmen by Georges Bizet.
This is a rather famous tune which I'm sure you've all heard. What can I say? I like it.
Can you be more specific? Symphonies or ensembles or solo piano, happy or sad, what period, etc?