The name of this potion is clever.
"unction" Eng. an ointment, derived from a Latin term meaning "to anoint with oil"
"unctuous" is derived figuratively from "unction" and refers to being flattering and friendly in an oily, slimy sort of way.
Effect: persuades the drinker that the giver is his/her very best friend
To allow himself to survive in Quirrell's body, Voldemort induced Quirrell to kill unicorns and drink their blood. Unicorn blood will give the drinker only a cursed life because the drinker has killed something defenseless and pure has been killed only to save his or her own life. Voldemort didn't care, of course, because he was only trying to survive until he could drink the Elixir of Life (PS15).
See Voldemort's potions (unicorn blood and snake venom mix).
Slughorn hoped to sell some unicorn tail hairs Hagrid had given him for 10 Galleons a hair. Hagrid collects such shed hair as a matter of course, using it as a binding on bandages of injured animals due to its strength (HBP22)