Slang term in the U.S. for a seat in an arena or stadium which is very far away from the action. Most of the times, these seats (also called the cheap seats, for obvious reasons) are positioned at ridiculous altitudes, on very steep slopes so that you can look down and see something besides the head of the person in front of you. They are named because the high altitude's thin air might encourage nosebleeds, but that of course is an exaggeration.
Most of the time, these are called "Upper Reserve," "Tier Reserve," or other such euphemisms. At Yankee Stadium during the 2002 season, one nosebleed seat ticket will cost you $17.00. Bring your own binoculars.
The nice part about the nosebleed seats is that during games which are not sold out, you get a lot of personal space. Very rarely, a foul ball will hit the nosebleed seats, prompting a mad dash to grab it. Aside from that, enjoy the game, don't get too drunk, and wait until at least the seventh inning to sneak down to better seats.