I use to have an account at the Maui supercomputer facility and did some work on one of the "grand challenge" type problems. I have not worked on a real big machine in 7 or 8 years now so a lot has changed.
The big thing is to simulate things like weather as well as complex defense problems. The reason they can not predicted exactly where a hurricane will go is because they do not have the computational power to have a ultra fine naiver stokes grid of the entire atmosphere (or do a direct simulation). That is, the storm comes and goes in real time before you answered the question of where it will go...so what is the use? So in order to have an approximate guess..the grid is made course, and other approximations are made introducing inaccuracies. The true mathematical problem size is immense almost beyond comprehension. The amount of FLoating Point Operations to be done is mind boggling. So, the approach is to get the fastest floating point processor you can and hook up as many as possible together along with as much of the fastest memory all coupled with the slickest interconnects and then try to tackle these. It is not surprising that the graphics companies are helping out here because their hardware has to compute alot of flops to display those smooth 3D graphics. Their problem was they were connected to a (relatively) small amount dedicated memory... I guess they have found a better interconnect to address more memory. Also you need really slick software to communicate activities between thousands of CPU's working on the same problem simultaneously.
What is cool is to see the solution, it is very satisfying to see a computational result from one of these bad boys. What is awesome is to give it a problem that your ultra fast dell workstation takes several days to solve...this thing spits out the solution is seconds.
There are other big machines (like the ones NSA has) that can listen to every phone call in the world and pick out key phases. But they are not solving mathematic nighmares . They are more high speed integer based machines with fast I/O. Big machines like this help decode DNA problems and alike.
The list changes every few months. Having the fastest is kind of big deal...The US should maintain it's leadership in this area. Usually NSA and a military funded site are at the top of the heap.
IMO Obama has been a disappointment with regard to R&D. Where is this "green economy" and all the new initiatives that were going to happen the day after he took office?
In my opinion he has taken his foot off the R&D gas and now others have caught up and passed us. It is sad to see us fall behind in this technology because it can lead to many other discoveries. We can take this back, we have the companies and the people who can do it.