This is a measure of the actual bitterness of a beer as contributed by the alpha acid from hops. Because the apparent bitterness of a beer is subjective to the taste of the drinker and the balancing malt sweetness of the beer this is not always an accurate measure of the "hoppiness" of a beer. But, generally speaking, beers with IBUs of less than 20 have little to no apparent hops presence.
Beers with IBUs from 20 to 45 are the most common and have mild to pronounced hops presence. Beers with IBUs greater than 45 are heavily hopped and can be quite bitter.
Examples: Not knowing that the barleywine had an IBU of 68, Rachel took a big swig from the glass then twisted up her face as the hops assaulted her taste buds.