1.x32 systems were once desired because they address (point to) 4 gigabytes in memory in one go. Modern applications require more than 4GB of RAM to complete their tasks so x64 systems are becoming more attractive because they can potentially address up to 4 billion times that many locations. If you have however a 64 bit only processor you cannot run a x32 OS on that system, and that some 32 bit programs might work on a x64 OS.
In computer architecture 64-bit computing is the use of processors that have datapath widths, interger size and memory adress widths of 64 bits (8 octets). Also 64 bit CPU and ALU architectures are those that are based on registers, address buses or data buses of that size. From the software perspective x64 computing means the use of code with 64-bit virtual memory addresses. Many programs are now available in x64. If you want to know what type your running? Open Conrol Panel and click System there it will show what type you are running.
Technically what he said ^ is one of the reasons why 64 bit computing is needed, in increases the linear space, systems such as Windows XP cannot address more than 4GB ram 2^32, although that was increased by using a 'hack' on 32 bit systems of the following operating systems afterwards such as windows 7 to support 2^36 space but the difference is the size of the instructions and as you 'know' about processors, i guess you know that it won't work if you try sending 64 bit instructions to a 32 bit processor. Then the instructions themselves depend on different architectures etc ..
However since Intel and such are all about compatability, the opposite, having a 32 bit system on a 64 bit cpu architecture is supported
Most current 64-bit processors can run either 32-bit or 64-bit operating systems. As noted, a 32-bit one can only handle about 4GB of RAM whereas a 64-bit one can handle about 4 billion times that (in theory). In any case, you should run a 64-bit OS on a 64-bit processor. Most can happily run either 32-bit or 64-bit applications equally well. This includes 64-bit Windows as well as Linux/Unix systems.