IBU is an International Bittering Unit, and is a scientific measurement that can be determined chemically in a laboratory. In the most practical terms, IBUs provide a scale from 0 to 100 of beer bitterness based on the additions of hops, and when and how the hops were added. IBUs can be measured in unfermented wort, but realize that beer will lose IBUs during fermentation, and over time, so while you can start with wort with IBUs much higher than 100 (yes this is possible), in the finished beer it might fall to say 80 IBUs, for example, by the time the beer is ready to drink, such that the upper limit truly is around 90 to 100 IBUs in any beer that is ready to drink, and for most beer styles will be much less.
Most homebrewers use a formula to estimate the IBUs, as we don't have easy access to a chemistry laboratory to actually measure IBUs. The most popular and accurate formula is known as the Tinseth calculation. People now understand that even Tinseth's formula isn't perfect. For instance it allows IBUs of 0 for beers that have much more than 0 IBUs, and >100 IBUs for beers that actually have only 60-80 IBUs or whatever. However, it is still just about the best we've been able to come up with, and truly is "good enough" for most intents and purposes.
Many beer styles will have as few as 10-20 IBUs, such as American light lagers and hefeweizens. Other styles can have IBUs approaching 100, such as IPAs. Most other styles fall someplace in between. A common average amount for an average beer is around 25-35 IBUs or thereabouts.
That's the short story. You can learn as much as you want all over the place in books and online. Many brewers devote a ton of time into understanding what hops do to beer, and how, including how the IBU is determined, and whether it matters, etc. It's an intriguing topic for many.