Galaxy collisions, protein folding, mechanical design and much more. Big simulations of real world physics and chemistry that require massively parallel computation, problems that require insane numbers of calculations running on multiple machines that can pass data to each other very quickly.

Every supercomputing center has a website where you can read about the research being done on their machines.

No, these can’t be done at similar scale on your desktop. Your PC can’t do that many calculations in a reasonable time. Even on supercomputers, they can take weeks.