I would name full albums, but I would imagine there won't be room, so I'll try to stick to songs (because picking albums isn't hard enough right?). I'm terrible at picking favorites, so I'm just going to put down a lot. Anyways, here goes.
John Frusciante - Former Red Hot Chili Peppers guitarist. Mostly rock and experimental music. Songs I recommend: Unreachable, Central, A Doubt, The World's Edge, Look On, Dissolve, Ratiug.
Fugazi - Pioneers of post-hardcore band. Solid all-around, ending with The Argument, one of my personal favorite records (I often use "Cash Out" from it as a reference. Songs: Cash Out, Sieve-Fisted Find, Shut the Door, Strangelight, KYEO
Pavement - Well-known indie rock band from the 90's. Some music credits have gone as far as to say the best band of the 90's. Songs: Father to a Sister of Thought, Grounded, Gold Soundz, Range Life
Stephen Malkmus & the Jicks - Stephen Malkmus of Pavement's current project. Songs: Cold Son, Stick Figures in Love, Houston Hades
Sonic Youth - Well-known alternative rock band. Songs: Teenage Riot, Mote, Kool Thing, Bull in the Heather
Pixies - Can you tell I like alternative/indie/punk rock? Songs: Gigantic, Where is my Mind, Debaser, Is She Weird?, Velouria
The Breeders - Alternative rock band fronted by Kim Deal of the Pixies and her sister. Most known for Cannonball. Songs: No Aloha, Drivin' on 9, Divine Hammer, Cannonball
Nirvana - I always find Nevermind at audio shows, which is their most sucessful album, but it's also my least favorite (same goes for many friends of mine who are Nirvana friends), not that it's a bad album or anything. Songs: Heart-Shaped Box, Dumb, All Apologies, About a Girl, Smells Like Teen Spirit (because everyone knows it)
The Replacements - Started as a hardcore punk band, and gradually shifted into one of the pioneers of alternative rock. Songs: I Will Dare, Unsatisfied, Here Comes a Regular, Swingin' Party, Hold My Life, Bastards of Young, Alex Chilton, The Ledge
Wilco - Well-known alternative rock/country band. Songs: At Least That's What You Said, Impossible Germany (one of my favorite guitar solos), Dawned on Me, Ashes of American Flags, Jesus Etc.
My Bloody Valentine - Pioneers of shoegaze. Songs: Only Shallow, When You Sleep, I Only Said
Blonde Redhead - One of my favorite indie rock bands. Songs: Elephant Woman, Misery is a Butterfly, Falling Man, Maddening Cloud, Melody of Certain Three, This is Not, SW, Bipolar
Radiohead - Everyone knows Radiohead. What more needs to be said? Songs: Fake Plastic Trees, My Iron Lung, Pyramid Song, You and Whose Army?, Paranoid Android, The National Anthem
Belle & Sebastian - Great indie pop band (though I found their last record boring). Songs: The State I Am In, I Want the World to Stop, Asleep on a Sunbeam, Roy Walker, Stay Loose, We Are the Sleepyheads, I'm A Cucuo, Funny Little Frog
David Bowie - David Bowie = God. It's that simple. Songs: Modern Love, China Girl, Young Americans, Space Oddity, Five Years, Sould Love, Moonage Daydream, Starman, It Ain't Easy
Neil Young - No introduction necessary. Songs: Hey Hey, Heart of Gold, After the Goldrush, Old Man, Helpless
Joni Mitchell - Another no introduction necessary. Songs: All of Blue. I can't pick from that album, the whole thing is perfect. Coyote off of Heijira would be nice as well.
Deerhunter - Newer shoegaze band. Songs: Desire Lines, Helicopter, He Would Have Lauged
St. Vincent - Art rock artist. Probably my favorite guitarist of recent years. All of her albums have been excellent. Songs: Now. Now., Marry Me, Marrow, Cruel, Surgeon, Strange Mercy, Dilettante, Digital Witness, Birth in Reverse
Jeff Buckley - Alternative artist known mostly for his cover of Hallejujah. Songs: Mojo Pin, Grace, Last Goodbye, Hallejujah, Lover You Should've Come Over, Dream Brother
Portishead - Pioneers of trip-hop, with heavy jazz influence. Their live album Roseland NYC is one of my favorite live albums (I must've watched the DVD 50 times already). Songs: Mysterions (live version from Roseland NYC), Glory Box (live version from Roseland NYC), Roads (live version from Roseland NYC), Strangers (live version from Roseland NYC), It Could Be Sweet, Wandering Star, Undenied, Only You, Western Eyes, The Rip, Small, Magic Doors, Threads
Broadcast - Really good indie electronic band whose singer died way too young of pneumonia. Songs: Come on Let's Go, Pendulum, Black Cat
The Velvet Underground - No introduction necessary. Songs: Sister Ray, Candy Says, I'm Waiting for the Man, Sweet Jane, Pale Blue Eyes, Rock and Roll
Kraftwerk - One of the earliest electronic acts. Songs: The Robots
Dinosaur Jr. - Indie rock band with one of the best guitarists in the genre. Songs: Feel the Pain, Watch the Corners, Start Choppin', Over It
Björk - I don't even know what genre she is, but she's got all sorts of stuff. Songs: Hyperballad, Jóga, Army of Me, Enjoy, Pagan Poetry, Hunter, Hidden Place, It's Not Up to You
Sleater-Kinney - One of the best bands to come out of the riot grrrl feminist punk movement. Man am I glad they're back, after a decade, and their new album is great. Songs: Dig Me Out, The Drama You've Been Craving, Start Together, Far Away, The Remainder, Light Rail Coyote, Sympathy, Jumpers, Entertain, No Cities to Love, No Anthems.
Outkast - No introduction necessary. Songs: ATLiens, B.O.B., So Fresh, So Clean, Ms. Jackson, The Way You Move, Roses
A Tribe Called Quest - Jazz rap group. Songs: Excursions, Verses from the Abstract
Killer Mike - Southern rapper, and a member of Run the Jewels. Songs: Untitled, Reagan
Run The Jewels - Excellent hip-hop duo. Songs: Get It, Oh My Darling Don't Cry, All My Life
Kanye West - No introduction necessary. Songs: Bound 2, All Falls Down, Homecoming, All of the Lights, Slow Jamz
Kendrick Lamar - Possibly the best rapper in the game right now. Last two albums have been insanely good. Songs: m.A.A.d city, Compton, Wesley's Theory, King Kunta
Alright...that's it. This has taken long enough. I obviously would only expect a tiny fraction of this to be considered, but to anyone that actually reads through this, thanks!