The bittering acid (alpha acid) in hops is naturally insoluble. Boiling the hops isomerizes the alpha acid into a form that's soluble, and will therefore add bitterness to the beer. And of course it won't all isomerize at once, it takes some time. Hence the need to boil for 60 minutes. Though you'll get some more bittering from boils longer then 60 minutes, the peak of the utilization curve is at just about 60 minutes, so there's sort of a diminishing returns situation.
Hop bittering is non-volitile, meaning they do not evaporate during the course of the boil.
Flavor compounds are somewhat volatile. Most of them evaporate in a 60 minute boil. But many will remain in a 30 minute boil. Less boil time means more flavor compounds remain in the beer. Flavor additions are usually between 30 and 10 minutes.
Aroma compounds are the most volatile. Boiling for 15 minutes will cause almost all of the aroma compounds to evaporate. Aroma additions are generally 10-0 minutes or dry hopped.
There is some overlap between the three. Even a 60 minute boil will not remove all of the flavor compounds; and boiling your flavor addition for 20 minutes will isomerize some of the bittering compounds. And likewise, late aroma additions will add some flavor and bittering, with the exception of dry hopping which is done at storage temperature and therefore adds no bittering.