What a great thread. My leanings will probably be a little different than previous posters. In no particular order (don't make me choose!):

Beastie Boys, Hello Nasty: Paul's Boutique is probably the better album, but I remember trading for this CD in middle school and not taking it out of my Discman for probably 6 months. To this day I can still sing most of the words and I use the same garble as I did at 13 for the lyrics I'm not sure of.
Beastie Boys - Hello Nasty (1/5) - YouTube

Bela Fleck, Bela Fleck and the Flecktones: Someone gave me this album for inspiration when I first started to learn the bass guitar, and it almost made me put the bass down forever. Victor Wooten flat wears it out on this album, and the rest of the players are elite in their own right.
Sinister Minister (Live from Bonnaroo 2011) - YouTube

The Black Crowes, Shake Your Moneymaker: I'm a child of the '90s, and this was one of the first "grownup" albums I can remember hearing. I also remember how pumped I was when I learned the intro to "She Talks to Angels" on the guitar when I was learning to play.
black crowes - she talks to angels - YouTube

City and Colour, Bring Me Your Love: Here's my sentimental side coming out. This was a favorite album of a girl I was in love with in college, and we listened to it on long car rides holding hands and stopping in parks to make out a little. That's as much of a relationship as we ever had, but it was fun while it lasted.
City and Colour- The Girl (Music Video) - YouTube

D'Angelo, Voodoo: I was so excited to hear he's making a comeback. This man is one of the baddest ever to do it, and if it weren't for his drug problem, i truly believe he would have 3-5 of the best neo-soul albums around. Plus, Pino is his bassist, so... This song is the funkiest thing I've ever heard.
D'Angelo - Chicken Grease (Live) - YouTube

The Decemberists, The Crane Wife: Not much to say about this album except that it's pretty close to perfection. These guys are getting better as they go along, but I don't know if they'll ever top this for me.
The Decemberists - The Island: Come and See (Live) - YouTube

Jimmy Eat World, Clarity: One of my good high school friends passed away when I was in college, and listening to this record makes it feel like he's sitting next to me in the car. I have to make plans to listen, because I almost always sit through it beginning to end.
Jimmy Eat World - Lucky Denver Mint - YouTube

Ryan Adams, Heartbreaker: My favorite album of all time by my favorite artist of all time. It comes from a really down time in his life, so lots of the songs he refuses to play live anymore. There's nothing like a fall day and this record together.
Ryan Adams w. Laura Marling - Oh my sweet Carolina - YouTube

Derek Webb, I See Things Upside Down: This is an interesting album by a Christian guy about life, love, God, all kinds of subjects. He used to be a folksy acoustic guitarist, which you still hear in this album, but he moves on to more electronic music after this. He has lots of good stuff, but this is tops for me.
Derek Webb - Medication - YouTube

Outkast, Stankonia: Good, trunk-rattlin' fun from one of hip-hop's most innovative groups. Remember when all rap albums had three or four skit tracks? Strong list of singles, and some of the lesser tracks (Humble Mumble being my favorite) are a blast, too.
OutKast - Humble Mumble - YouTube

Matchbox Twenty, Yourself or Someone Like You: I don't care what anyone thinks; I loved '90s pop/rock music. This album (and their next, Mad Season) was the soundtrack of the late '90s and had like six singles on it. Really easy to listen to.
Matchbox Twenty - Real World (Video) - YouTube

Van Morrison, Astral Weeks: Someone else mentioned this record above, and I completely agree that it's the best one to come out of the '60s. "The Way Young Lovers Do" is certifiably brilliant, and the title track is just gorgeous. This album was WAY ahead of its time while remaining classic.
Van Morrison - The Way Young Lovers Do (live at the Hollywood Bowl, 2008) - YouTube

Incubus, Morning View: This is an album that changed the way I listened to music. I remember being fascinated by some of the rhythms and hooks and the way Boyd sometimes seemed only passingly interested in syncing his vocal phrasings with the track itself. "Wish You Were Here" has one of the great simple choruses.
Incubus - Nice To Know You - YouTube

Nickel Creek, Why Should the Fire Die?: This is another one of my favorite bands ever, and while their "newgrass" music has always been lovely, the rock n roll turn they took on this one really cemented their genius for me. I wore this album out in college, and even bought a mandolin to try and learn Thile's parts. I made it as far as "When In Rome" before realizing it wasn't going to happen. I'll always have that one song in my repertoire, though.
Nickel Creek - When In Rome - YouTube

Garth Brooks, Double Live: That's right--I'm putting country music on here! I've never been a big country fan, but Garth Brooks has to be one of the best songwriters in country/pop music. And I'd put his live shows up against anyone. Double Live (part of which was recorded in Rupp Arena) brings out that energy, and I've spent many a long drive belting out every word to the double album.
(Sorry, can't find a video. Some of you are probably happy about that.)

Michael Jackson, Thriller: It's really hard for me to pick one record from MJ, but I settled on Thriller because it has two of my three or four favorite MJ songs ("PYT" and "Human Nature") and includes some of the most iconic.
Michael Jackson - P.Y.T. (Pretty Young Thing) - YouTube

Over the Rhine, Drunkard's Prayer: This is probably the slowest album on the list and my favorite from the local duo. The first band I ever played in covered "Born," and I've been hooked ever since. Her voice is just so sweet, and it's always a good pick-me-up if I'm a little down.
Over the Rhine - Born - YouTube

Jay-Z, The Black Album: I had never been a big Jay-Z fan until this album came out, and I became a believer. When I got my first car, this is one of the albums that I played on my first drive around. It was an old, beat up Mercedes. I loved that car.
Jay - Z - Encore - YouTube

Mumford and Sons, Sigh No More: I know this record got played out for a lot of people, but I remember exactly where I was the first time I heard it. I was standing in my friend's NYC apartment in December 2009, the first time I'd ever been to what became my favorite city in the world. I remember thinking it sounded like New York but also like my home in Kentucky, and Ryan Adams was the only other artist that gave me that sensation.
Mumford & Sons - Awake My Soul (Live) - YouTube

Prince, Purple Rain: There's never been anyone like him, and there never will be again. "When Doves Cry is actually the song that I knew from dancing to it as a little kid, and years later I discovered (rediscovered) the title track, which is stunning. Between the guitar solos and the woooos, it's a transcendent song.