Tom Parke, posting in rec.arts.dance, offered the following definitions:
If the dancers are attempting to prove that gravity does not exist, then it's ballet.
If the dancers are attempting to demonstrate that gravity does exist and it's a bitch, then it's modern.
If the dancers are attempting to demonstrate that gravity does exist but they'd rather die fighting it than give in to it, then it's jazz.
8)
I think of Modern Dance as a down and dirty form of ballet. They follow a lot of the same principles but are willing to show the hard side of life. Ballet comes from aristocratic dancing and is very polished and, at its best, tries to defy gravity as stated above. They work very hard to look as if they are lighter than air. Modern dancers work very hard to show you the air they are dancing through, over, and under.
Jazz Dance comes from a totally different viewpoint- first and foremost - the music. Jazz dance developed from Jazz and Swing music. It is used most often in Broadway and Movie Musicals.
Here are probably more links than you want to check out, but they will give you a good overview of the styles. I did a Google check on Jazz dance and Modern Dance and surfed from there.
Good short history of Modern Dance
http://www.factmonster.com/ce6/ent/A0833540.html
Photos of Modern Dance
http://www.duke.edu/~saundra/
Good short history of Jazz Dance
http://www.danceart.com/gotjazz/whyjazz.htm
Discussion of Modern Dance & Ballet
http://www.panix.com/~twp/dance/faq_1.htm
Discussion of Jazz Dance and developers of technique
http://www.dance-teacher.com/backissues/aug00/jazzdance.shtml
Jazz technique developers
Luigi
http://www.luigijazz.com/
Gus Giordano
http://www.giordanojazzdance.com/
Jazz Dance Choreographers
Bob Fosse
http://www.nodanw.com/biographies/bob_fosse.htm
Jerome Robbins
http://jeromerobbins.org/home0.htm
He was trained in ballet, has modern influences, and choreographed West Side Story with strong Jazz styles.
and sites for some of the better known modern dance choreographers
Martha Graham
http://www.marthagrahamdance.org/
http://www.time.com/time/time100/artists/profile/graham.html
Merce Cunningham
http://www.merce.org/
http://www.pbs.org/wnet/americanmasters/database/cunningham_m.html
Alvin Ailey
http://www.alvinailey.org/
Paul Taylor Dance Company
http://www.ptdc.org/