I am trying to learn the definition of "josefiterne samfund". This is
some kind of religious group with which my great grandfather travelled
around the north part of the Jutland peninsula in Denmark  from about
1878-1880. I have already checked the records of the LDS (Mormon)
church, and the Reorganized LDS or now named Community of Christ
Church without success. Can anyone tell me what the Josefiterne
Samfund was and who has  the records for that organization from
1878-1880?

It seems that you're on the right track in looking to the Reorganized
LDS, now the Community of Christ.  So far as I can tell, "josefiterne
samfund" means Josephite society or community, which may be an
unofficial designation for this group.  (It's also conceivable that it
refers not to the Reorganized LDS, but to Catholic Josephites.)

You can see a reference to Josephites in Denmark in the 1880s on this page:

"Misc. Pennsylvania Newspapers: 1850-1899 Articles" ("Old Mormons," April 13, 1887)
Uncle Dale's Old Mormon Articles
http://www.lavazone2.com/dbroadhu/PA/penn1860.htm

Perhaps someone at the Europe Mission Center of the Community of
Christ could provide assistance.

"Church Directory: Europe"
Community of Christ
http://www.cofchrist.org/directory/world/europe.asp