First, Ill talk about the pure mathematics and structure of the processors that are involved here. Ill keep this part short and sweet.
A bit is short for binary digit. It is basically how a computer stores and makes references to data, memory, etc. A bit can have a value of 1 or 0, thats it. So binary code is streams of 1s and 0s, such as this random sequence 100100100111. These bits are also how your processor does calculations. By using 32 bits your processor can represent numbers from 0 to 4,294,967,295 while a 64-bit machine can represent numbers from 0 to 18,446,744,073,709,551,615. Obviously this means your computer can do math with larger numbers, and be more efficient with smaller numbers.
Now see, that description wasnt too bad, but the question is how does this affect you, the average PC owner? The largest benefit will go to academic institutions and private companies, where large calculations are being performed, huge databases are being accessed, and complex problems are being solved.
Everyone that doesnt fall into that category will see some benefit of using 64 bit processors over 32 bit processors, but not much in todays marketplace. The AMD Athlon 64-bit processor is completely backward compatible, meaning you can currently use it with 32-bit operating systems and software programs. You will see some benefits by using this setup, but because the programs werent written to take advantage of the extra power, they wont use much of it.
The true benefits of this set up dont come from the amount of bits, but by the improved structure of the 64 bit vs 32 bit processor's older structure. A 64-bit processor is made with more advanced silicon processes, have more transistors, and faster speeds. This is currently where the true benefit of switching to a 64-bit processor lays.
As for 64-bit operating systems and software, many are in the works, but nothing is in final version. Microsoft has released a beta version of Windows XP that takes advantage of the 64 bit technology, but there are still issues. The problem is when you run 32-bit software programs in the environment of a 64-bit operating system. Many programs wont work properly, such as Adobe Acrobat and Windows Media Player, for example. Another issue is RAM. You really need about 4 GB of RAM to take full advantage of the capabilities offered by a 64-bit processor, while most PC owners have less than 1 GB under their computers hood.