Nicholas St. North
Here I'm starting with a part from the book, that in my opinion defines the relationship between North and Bunny, in both mediums: book and movie.
Katherine sighed. Here they were, the oldest and wisest creature on Earth and the greatest warrior-wizard of the age, yet they were behaving like a pair of brats.
- p.195 in E. Aster Bunnymund and the Warrior Eggs at the Earth's Core!
For the movie the creators admitted they wanted to make a Kirk/Spock relationship between the two, but let's face it, it couldn't work out, not because at how movie Bunny came out, but even because at what kind of wacko are Bunny and North in the books.
I would define this relationship as best friends, the two love to bicker, but at the same time, when the things are serious they are able to work together as a well oiled machine.
In the animated version it's shown that the major arguments between Bunny and North are about the fact that Christmas is more important than Easter, and that's something that Bunnymund doesn't likes. But even North can admit when it comes to children believing, the Easter in the movie is the one that is more important than any Christmas, because they are putting at risk children's believe.
North: “As much as it pains me to say it, this time Easter is more important
Bunny: "Hey did everyone hear that!"