The Black Ears are one of the hill tribes that live in the Mountains of the Moon. They derive their name from their habit of chopping off the ears of their victims and wearing those as a necklace. They are ruled by Chella, daughter of Cheyk.
The Black Ears are among the hill tribes that Tyrion recruits, in return for money and weapons to take over the Vale for themselves. They accompany Tyrion to his father's military camp, and he agrees to honor his son's promises to them.
In the books
While the Black Ears do cut the ears off of their fallen enemies, they don't necessarily kill them. As Chella explained: "Only cowards kill the vanquished; braver to leave the man alive, with a chance to cleanse his shame by winning back his ear; only so can you prove you do not fear your enemies."
Chella and more of the Black Ears are among the clansmen who follow Tyrion to the Lannister camp, and subsequently participate the battle of the Green Fork.
Prior to the battle of the Blackwater, Tyrion recognizes that the Black Ears and the other hill tribes, while ferocious, are undisciplined and inexperienced at siege-warfare, so he sends them south across the river to the Kingswood to harass the scouts and raid the baggage train of Stannis's army as they approached the city. They perform quite well, destroying Stannis's scouts and screening forces to such an extent that he has no warning of the approach of Tywin's massive reinforcements approaching the city.
In "A Storm of Swords", Bronn informs Tyrion that Chella turned up with a dozen Black Ears at the River Gate one morning, but the Lannisters guards chased them off while the Kingslanders threw dung and cheered. Tyrion is enraged at that ingratitude.