Wolves, Lower - R.E.M.
And then later, Radio Free Europe (also R.E.M.)
The songs were dark and the lyrics were marvelously inscrutable, and then there was the jangly Peter Buck guitar work, and I was thunderstruck.
My list of songs that hit me the first time I heard them, though not all of them changed my life:
Every Day is Like Sunday by the Smiths - pretty ironic, I got into the Smiths just as they were breaking up. I’ve been a fan ever since. I saw Morrissey live a few weeks ago, and just about died when he sang How Soon is Now, another song for this list.
Everybody Knows by Leonard Cohen. The original, not the Concrete Blonde cover. The song that made me a Leonard Cohen fan.
Sheela-Na-Gig by PJ Harvey. I was in a situation pretty much like the one in the song at the time I first heard it. Finding out what a sheela-na-gig is did not diminish my enjoyment of the song.
Kumbala by Maldita Vecindad. I guess because it evokes the atmosphere of a dimly lit nightclub, a couple (feeling as if they are) all alone on the dance floor, and the feeling that anything can happen.
Lately, the song that hit me and I can’t get enough of is Into the Night by Santana and Chad Kroeger. I don’t know why, maybe for the same reason as the above song. This song is faster and more cheerful though. I need to get out more.
My mother was a country music fan, so that was what I grew up listening to. One day when I was about 10 or so, I was fooling around with the radio and happened upon David Bowie’s Space Oddity.
I have never been the same.
Yes, I was born in 1960, why do you ask?
I won’t even pretend to understand the lyrics of modern music, but I don’t think you’ll get the answers in this thread. Songs that hit me as wonderful right away do so because of the music and the overall feel, and I’ll get to liking the lyrics after I’ve heard it four or five times.
The Way by Fastball. Great sound, and I had a moment of “Who is this?” when I first heard it.
Let Her Cry by Hootie and the Blowfish. Yeah, so I was too young to remember the 90s stuff and rediscovered what I should have been listening to when I was four. Got a problem with that? 
*Nightmares by the Sea * (Jeff Buckley) and *She Sells Sanctuary * (The Cult) come to mind. *Nevada, California * (Jayhawks) also. A few others.
Great lists by all, I will add.
Slight nitpick - solo Morrisey, not Smiths.
And thanks for the compliment, Sapo!
Cocteau Twins’ Blue Bell Knoll - Unintelligible glossolalia lyrics, ethereal guitar music, angelic voice.
Joy Division’s Atmosphere - End of love, end of life, sadness, loneliness, existential Angst in the truest sense.
Dead Can Dance’s Saltarello - a modern rendition of a medieval dance, no lyrics.
Shelleyan Orphan’s Shatter - Lyrics don’t make a lot of sense - about Angst, again, but it’s all about the music and voice.
My Bloody Valentine’s Strawberry Wine - a love song, shoegazey guitars
Sugarcubes’ Birthday - A lovely ditty about a little girl’s birthday with paedophilic subtext 
Suffragette City.
When I was 12 and when he got to “wham bam, thank you mame” I discovered sex. Well not really, I’d been reading my aunt’s Penthouse for years, but I had an epiphany and literally fell over backwards (fortunatly I was sitting at the time).
Interestingly enough, the song was inspired by a story similar to what is described in the song. And elderly couple, the wife with Alzheimer’s and the husband recently having brain surgery, left for a family reunion but never arrived. The song gives them a “happy ending,” of sorts, saying they meant to just drive off and leave it all, as opposed to them not knowing what they were doing and dying.
U2: Running to Stand Still from Rattle and Hum
Jeff Buckley’s cover of Leonard Cohen’s Hallelujuh. I first heard it on an episode of West Wing, and that soaring voice had me awe-struck. I felt impaled on its beauty…I think I forgot to breathe for a bit.
Madreblu’s “Certamente”
Dire Straits “Sultans of Swing”. Loved it from the first time I heard the intro played on the radio.
I was just thinking about this the other day. I have several that just absolutely floored me the first time I heard them:
*Under Pressure * by David Bowie and Queen
Let’s Dance by David Bowie – I was at the Mudd Club in NYC for a concert. During the pre-show music, this came on and I totally freaked !
I Will Follow by U2 – Same story, different club.
Science Fiction Double Feature from RHPS
so many…
How Soon is Now? The Smiths - the swirling guitar and droning lyrics were totally unique and stopped me dead on the dancefloor in college
Pain Lies on the Riverside Live - I still remember hearing this for the first time and feeling the energy and power of the music… and they were only 17…
Once Pearl Jam - I was waiting for something new in the early 90s and this was it
Dear God XTC - I always loved Andy Partridge and there were songs on every XTC album I loved, but this song stunned me… it said everything…
Smells Like Teen Spirit Nirvana - I had never heard anything like this… it made me stand up and get moving…
Freshman The Verve Pipe - totally haunting - was sorry they never “made it” but this song hit a spot in me…
A Sort of Homecoming U2 - maybe because my girlfriend was overseas at the time, but the “separation, dislocation… let it go” was like a sledgehammer…
Tim McGraw, “Live Like You Were Dying.” I realized I wasn’t so young anymore and it was time to get going on a few things I’d always planned to.
John Lennon, “Beautiful Boy.” Maybe not the entire song, but certainly the one line, “Life is what happens when you’re busy making other plans.”
Van Morrison, “Moondance.” My first introduction to Van Morrison, and it made me curious to hear his other music.
Queen, “Bohemian Rhapsody.” Just because it stood out among the other music of the time.
Klaatu, “Sub Rosa Subway.” Extremely complex, production-wise, and easy to see from this song how the “Klaatu is the Beatles reunited” rumours began.
I remember the first time I heard Simon & Garfunkel performing “The Sounds of Silence.” It was September, 1965. I had the radio on, and was barely listening while I knitted a scarf. The song made me set aside my knitting, and I called the radio station to ask about the song and the singers. Apparently quite a few people did the same thing, because I got a busy signal from the radio station’s phone line.
from my earliest memories of music and romance:
Magic - Pilot (you remember: Ho, ho, ho It’s magic, you kno-o-ow)
Hooked on a Feeling - B.J Thomas? - coincided with my first crush in 3rd grade
My Eyes Adored you - Frank Valli - just because…
Here Comes the Sun - George Harrison - writing notes to my 7th grade girlfriend using his metaphors, thinking I might actually get “some” even though I didn’t know what “it” was
Come Sail Away - Styx - this was almost as hard to slow dance to as Stairway to Heaven, but i recall many an adolescent boner dancing to this one at record hops…
jumping a decade forward:
Ten Feet Tall - XTC - riiiiight, the chemistry is riiiight…
I Love You - Translator - an obscure song from the band best known for “Everywhere that I’m Not”… I only have this song on an old cassette tape, but remember it being a part of my first college love
The Two of Us - The Beatles - “we’re going home” somehow became my college slang for heading back to the dorm to boff…
Crazy For You - Madonna - this was a frat party staple, and it moved all the girls in their rubber black bracelets and bras over their clothes…
damn… great thread… now I can’t stop thinking about the music from my life…
The full list would stretch on forever, but here’re a few recent ones:
Joe Strummer’s “Johnny Appleseed” had me watching John from Cincinnati for a couple episodes after I had given up on the show itself. Then I just bought the CD.
Bonnie Prince Billy’s “Strange Form of Life”
And a couple older ones:
Leonard Cohen’s “Famous Blue Raincoat”
R.E.M.'s “Country Feedback” and “E-Bow the Letter”
Elliott Smith’s unreleased cover of “Trouble” by Cat Stevens. You know, the song that plays at the end of Harold and Maude. That song was the last one he ever recorded.