Most massively parallel machines usually involve a processor with it's own memory interlinked with other processors using some common form of networking - like IEEE1394, etc. They do not run programs like a PC, the memory is usually shared across the entire computer (usually, but not always - it depends on the architecture). So, a machine with 2048 processors each with 16MB of RAM, essentially has the same amount of memory as one 16MB machine. This is noticeably different from the SMP multi-processor architecture of the PC.
While there are a lot of problems that are readily solved with massively parallel machines (finite element, matrix analysis, weather prediction, and problems that border on chaos theory like aerodynamics), there are plenty of other problems that are not adaptable to a massively parallel machine (like Unreal Tournament).
<< I would think that the only possible way to do it is to use the quantum characteristics of individual electrons right? >>
We are quite a ways away from this right now. Quantum computing - if you can even call it that - is still in it's infancy. They have barely even demonstrated "proof of concept" let alone prototype machines - deploying a massively parallel quantum computer is still well over a decade away... optimistically.
<< What I understand about the normal processors is that they only make calculations using two electron states, which are designated 0 and 1 right? >>
Conventional (non-quantum) computers are based entirely on binary arithmetic. Everything is based on zeros and ones, on and off, high voltage or low voltage. There are two levels and everything is based off of this. But they are not electron states. The charge holding data in even a 0.18um 6T memory cell involves millions of electrons (or probably more - I've never counted) - not just a single electron. I don't understand how, considering Heisenberg(sp?)'s theory and quantum statistics, even quantum computers are going to store data on one or two electrons.
One of the advantages of quantum computers is said to be the fact that they are quantized into more states than just two. So, rather than binary arithmetic, they might operate on hexadecimal arithmetic. Or perhaps even higher. This alone would result is vastly smaller and faster chips - even leaving aside the fact that these devices are built out of single atom devices. But still, this is a field that has barely even figured out how to define the devices, how to create circuits using them, and they still haven't figured out how to wire them up last time that I checked. They have a LONG way to go.
I don't mean to demean the work that is going on. It's fascinating, exciting and cutting edge work and I'm constantly amazed at what they have achieved. But make no mistake: they have a long way to go to even prove that this approach is practical (IMHO).