I have been trying to read up on 32-bit and 64-bit processors (http://en.wikipedia.org/wiki/32-bit_processing). My understanding is that a 32-bit processor (like x86) has registers 32-bits wide. I'm not sure what that means. So it has special "memory spaces" that can store integer values up to 2^32?
I don't want to sound stupid, but I have no idea about processors. I'm assuming 64-bits is, in general, better than 32-bits. Although my computer now (one year old, Win 7, Intel Atom) has a 32-bit processor.
All calculations take place in the registers. When you're adding (or subtracting, or whatever) variables together in your code, they get loaded from memory into the registers (if they're not already there, but while you can declare an infinite number of variables, the number of registers is limited). So, having larger registers allows you to perform "larger" calculations in the same time. Not that this size-difference matters so much in practice when it comes to regular programs (since at least I rarely manipulate values larger than 2^32), but that is how it works.
Also, certain registers are used as pointers into your memory space and hence limits the maximum amount of memory that you can reference. A 32-bit processor can only reference 2^32 bytes (which is about 4 GB of data). A 64-bit processor can manage a whole lot more obviously.
There are other consequences as well, but these are the two that comes to mind.