A nosebleed or bloody nose was a condition where the capillaries in one's nose burst, causing blood loss. It could be caused by an actual blow to the nose or manifest as a psychosomatic response. The term could also be used as colloquially to describe an embarrassing wound to one's pride.
In 2255 of the alternate reality, James T. Kirk suffered a nosebleed after getting into a fight with Hendorff and other Starfleet Academy cadets at the Shipyard Bar. Later, he used wadded-up napkins to stem the flow of blood. (Star Trek)
In 2371, Lupaza reminisced about the days in which the Shakaar resistance cell would give the Cardassians a bloody nose when they got too close; Shakaar Edon reminded her that sometimes their nose got bloodied. (DS9: "Shakaar")
In 2374, Jadzia Dax, assuming the role of Romulan Proconsul Neral for a hypothetical discussion with Benjamin Sisko, claimed that Dominion incursions into Romulan space to give the Federation a bloody nose didn’t make "her" feel bad. (DS9: "In the Pale Moonlight")