This is a list of acapella songs (voice only, no instruments) that I think are cool. I have put them into approximate order of favourite (scientific eh?). I think that it's pretty funny to see how many of these have been song of the day!:
PandemoniUM - No One Knows
MIT Resonance - Mr. Brightside
Oxford Out of the Blue - By The Way
UC Men's Octet - With or Without You
Oxford Out of the Blue - Let me entertain you
UC Men's Octet - Bohemian Rhapsody
UC Men's Octet - Accidentally in Love (with One Headlight slipped in there too)
The Harvard-Radcliffe Veritones - High and Dry
Soulstice A Cappella - Hide And Seek
UC Men's Octet - Under Pressure
PandemoniUM - Semi-charmed
Ryan Malcolm and Reagan Smith - Sound of Silence
Umass Dynamics - Such Great Heights
Oxford Out of the Blue - Fat Bottomed Girls
Reverse Osmosis - Take your Mama
Oxford Out of the Blue - Superstition
Reverse Osmosis - Sugar, We're Going Down
Carleton Singing Knights - God Only Knows
4-Tune Hunters - Come Together "This one is for all you women out there - you know who you are"
PandemoniIM - District sleeps alone
USC Reverse Osmosis - Knights of Cydonia/Carol of the Bells
USC Troy Tones - Push
Troy Tones - Teenage Dirtbag
Trojan Men - Don't Stop Me Now
Carleton Singing Knights - Chicago
UCSB BFOM - Eleanor Rigby
Williams College Ephlats - Unwell
Diapaswing - Video killed the radio star
The Undertones - Jack and Diane
BYU Noteworthy - Unwritten
The Testostertones - For The Longest Time
Carleton Singing Knights - Hello Goodbye
USC TroyTones - (I Just) Died in Your Arms
Whitman College Testostertones - Stacy's Mom
BYU Vocal Point - Thriller
Brown Derbies - Halleluja
The Harvard-Radcliffe Veritones - The Sweet Escape
UC Men's Octet - Don't Wanna Miss A Thing
The Harvard-Radcliffe Veritones - They
Cal Poly Take it SLO - Boulevard of Broken Dreams
The Carleton Singing Knights - Hook
UO On The Rocks - Smile Like You Mean It
UC Men's Octet - Wake me up before you go go
UCSB Naked Voices - Feelin' Good
UC Men's Octet - 500 Miles
Time Check - Fix You
UCB Men's Octet - Billie Jean
UCacapella - Gangsta's Paradise
Embry Riddle Acafellas - Breakfast at Tiffany's
The Undertones - Take On Me
Brown Derbies - Our Time is Running Out
Oxford Out of the Blue - Crazy Love
Ransom Notes - Killing Me Softly
Tongue-Tied - We Built This City
USC SoCal VoCals - You Give Love a Bad Name
UCLA Awaken A Cappella - Walking in Memphis
Embry Riddle Acafellas - Kissed by a Rose
The Penny Loafers - How to save a life
UCLA Awaken Acappella - Book of Love
PandemoniUM - S.O.S./Tainted Love
CSUN Acasola - Bittersweet Symphony
No Strings Attached - Never Ever
SoCal VoCals - Total Eclipse of the Heart
Reverse Osmosis - Flashdance (What A Feeling)

And for the comedy final one:
PwnStarr and the PwnTones - Take me out

If you've managed to watch all of those then you're possibly running out of YouTube - but you now try the following three megamixes (which include a several of the above songs):
WHS Fortune Hunters - Super Medley (adapted from the original by the Whitman College Testostertones)
Acasola - Techno / House Medley