The nose is part of the humanoid face that contains the organs of smell and marks the beginning of the respiratory tract.
According to Jonathan Archer, Tycho Brahe lost his nose over a math equation. (ENT: "The Andorian Incident")
The holographic character Nicky the Nose was named for his metal nose. (Star Trek: First Contact)
In 2371, B'Elanna Torres broke Joseph Carey's nose in three places after an argument in engineering. (VOY: "Parallax")
Beetle snuff is inhaled by way of the nose. (DS9: "The Nagus")
Worf developed a gorch on his nose. (Star Trek: Insurrection)
The Doctor was going to have a topic on Hints for Healthful Living called "How to keep your nostrils happy". (VOY: "Investigations")