This super-bad unicorn of the sea is most closely related to the beluga whale. The unicorn-horn is not a horn at all but a tooth that grows through the top lip. Male narwhals have one overgrown tooth (1 in 500 have 2) that usually comes from the left and in mega rare cases the female can have a tusk as well. The body length can be up to 3 meters and the tusk can be half that size. So, why do they have a tusk? Well, not for hunting because they eat benthic foods. Fighting? Breaking through the ice? These behaviors have rarely been observed.
There are two thoughts on the purpose of the tusk: secondary sexual characteristic or environmental barometer.
Secondary sexual characteristics may function in male to male combat in the case of horns or antlers and the narwhal tusk, if they do indeed use them for fighting. The tusk may also be an ornament like the mane of a lion or the feathers of the peacock. The long tusk could be the male narwhals way of telling the ladies that he is the business. In this manner, males can be seen rubbing their tusks against other tusks and this behavior may aid in establishing a dominance hierarchy.
In 2005, Dr. Martin Nweeia from the Harvard School of Dental Medicine reported that the tusk or tooth was used to sense conditions in the sea and in the air. When you consider that most of dental research must be extremely boring, this guy has found quite possibly the coolest question to wrangle.
This debate over the function of the narwhal horn is far from being resolved and if both sides are right (highly UNlikely), this would be an interesting case of a secondary sexual characteristic having a beneficial purpose outside of attracting mates.