Hi Marko,
Like Lugh said, HPC is a big area and can be performed in many ways. Depending on what you are trying to achieve you have different possibilities. For example, if you are looking to speed up Monte Carlo simulations, GPU might be an option. Other type of parallel programming pattern like MapReduce can help you to speed up certain king of processing. Then you have things like FPGA, or Clusters. When trying to speed up things, most of the time you have to adapt your programming style to the machine architecture. I know it is kind of generic as an answer, but it is simply because performance can be obtained is so many ways. I would say that it should always start with good programming habits.
Regarding the application to quant finance, HPC is used to speed up computing time in order to obtain the closest thing possible to instantaneous calculation. This is very used in HF trading, and derivatives pricing. In HF trading the need for HPC is on the full range of the application, so you have needs for data mining related operations, heavy computing algorithms, I/O, network. For derivative pricing speed is usually obtained through grid computing. There are probably more things to say that I am not aware of. From what I understand if you work as quant for a big IB, you don't need to know these things, as you will have a team dedicated to address these issues. You probably need to know more about it, if you plan to work for a smaller firm.
Hope this helps