in college basketball, for me it used to be BYU as they were the main enemies of my school UNLV
but now that BYU took its whiny butt out of the Mountain West Conference starting this year, I am not sure there is a team I really dislike anymore. I just want my teams (Seahawks and UNLV) to win no matter who they play.
Edit: I just realized this thread was in the NFL forum (I first saw it by looking at New Topics.) but BYU was really the only team in any sport I strongly disliked.
In all sports the only team I truly despise is the Yankees (my fantasy baseball team is Yankee Haters Inc.). In the NFL I enjoy rooting against the Cowboys, but its nowhere near the level of the Yanks.