When I was a teenager I didn't think there was much emotion in music. A few songs stirred feelings in me, but I thought people exaggerated when they said music made them cry. People said some music was emotional, but I either hated "emotional" music or it left me cold. Some music made me feel something, but nobody called it emotional so I didn't see it as emotional.
Then I bought Yes's Close to the Edge, one of the best albums ever. Most people have to warm up to it because it's weird and complex – I loved it the first time. I discovered something new in music. Close to the Edge has a huge scope of emotion, but they don't do it in typical ways. Nothing sounds like power ballads or adult contemporary. The lyrics are weird but they fit the music well. The crazy thing is a lot of people, including some "Yes fans", think their music is all logic and no heart. If you say you're a fan and you don't get any feeling from their music, you're missing a lot. Getting into Yes helped me enjoy other music more.
What triggers my emotions is different from other people. I don't care much about lyrics. I pay attention to lyrics. They can change my opinion of a song. But they come second to the music. To many people songs are lyrics with music attached. Also, blues music leave me cold. It uses the most neutral chord structures. The blues scale sounds neat but it also sounds neutral. I can enjoy blues with my head, but not with my heart. I don't like the DIY punk ethic either. I hate the idea that if you sing well or play well, it makes your music "fake". Bad singing/playing keeps me from getting emotion from music.
What triggers feelings in me? Great songwriting. Beautiful melodies are one of the best ways to get to me, especially with complex/dissonant harmonies. Good arrangements are another way. I like weirdness, but I hate weirdness for the sake of weirdness – it has to be natural weirdness. Great playing can take a good song over the top. I usually like complexity, but not always – some simple music is just as good. Tension and release dynamics are awesome. So is unpredictability. A lot of my favorites are mellower than most of what I like. A lot of my favorites are heavier than most of what I like. I get something from some sad music, but also from happy music, angry music, scary music, energetic music, peaceful music, and a lot of music that's hard to pin down.
Artists who are good at emotion include Yes, King Crimson, Procol Harum, The Beach Boys, The Beatles, (early) Chicago, Stevie Wonder, Dream Theater, Muse, House of Heroes, Eisley, Cocteau Twins, Brian Eno... Good genres for me include progressive rock, some hard rock/metal, some mood music/ambient, and the more complex side of pop. Some classical music works well too, especially the Baroque and Romantic eras.