Ears are, beyond being hearing organs, specific trophies for some of the more violent people of the wasteland.
If the Chosen One has around 110% in Unarmed, they can bite the Masticator's ear right off in the Jungle Gym's boxing ring. However, if one has less than 110 hit points, the Masticator can bite theirs off. The Masticator's ear is worth $750, but the Chosen's will cost a point of Charisma. These items are a reference to the infamous boxing incident when Mike Tyson bit off a portion of Evander Holyfield's ear.
Daniel Littlehorn from Littlehorn & Associates, located at the Scrapyard, rewards the Lone Wanderer with 15 caps and bad Karma for each ear. This is the bad Karma alternative to the Lawbringer perk, in which fingers are collected instead of ears.
While essential characters cannot be killed, an ear will spawn in their inventory each time they "die". These can then be pick-pocketed from them.
Every felled Caesar's legion soldier has a chance of having a legion ear on his corpse after starting the associated unmarked quest An Ear to the Ground.