The difference between 32-bit and 64-bit lays, basically, in how much RAM your processor/CPU (=brain of your computer) can handle. The more bits your CPU/processor can handle the more, quicker, efficient and accurate it will work 
32-bit processors can handle up to 4 GB of RAM and 64-bit processors can handle above 4 GB to 128 GB of RAM.
Yup, this means that in order to run a 64-bit version of CentOS you need to have at least 4 GB of Ram. Read this, if you are not sure if you can run 64-bit on your computer
In Linux (generally speaking), you need to run a 64-bit version of (for example) CentOS to run software/applications written for 64-bit systems. Wait, don’t give me the “like duh 
” reaction yet!
The good thing about having a 64-bit environment is; you will be able to run 32-bit software/applications too 
Unfortunately, it doesn’t work the other way round.
Guys, this is a very simplefied explanation. I’ve read articles regarding this subject which were so 1337, my brain exploded from just reading them 
La Boina Roja 
August 9, 2012 at 2:27 pm
Just a mild correction:
You can run 64-bit Linux (be it a Fedora descendant or anything else) with less than 4Gb of memory. You probably need >200Mb to make Gnome or KDE happy, though, and I think that the installer typically needs at least that much to succeed (which is odd, since a minimal system with no GUI requires only a few Mb to run).
Also, something not very well known about 32-bit processors is that nearly all of them since the 80’s actually have 34 to 36 bits of addressing space (whoa! everyone forgot about that!) so you can generally use up to 16Gb of RAM with a 32-bit processor, but you have to have the PAE addressing bits accessible from your kernel. This is the standard for Linux (who wants a crippled kernel?) but not for Windows, where Microsoft built support for >4Gb systems in, and then disabled that unless you pay for the “Pro” or better versions of their operating systems (and this whether or not it is a 32 or 64 bit version. This as true at least as recently as Vista, but honestly I haven’t used any Windows systems for quite a while so hopefully its not the case with Windows 7 or newer).
There are other details, as you are no doubt aware, but as far as RAM address space goes, the story isn’t nearly as straight forward as the marketing press would have you believe. This is particularly true of the 90% of the marketing press that was paid to push “Pro” versions of Vista back when the great 32 -> 64 bit transition was still a new idea for most people. Most articles about this are around 5 years old for this reason.
The bigger details that affect the Linux user base is multi-lib support. Like “what are all these i686 rpms doing on my system?” That’s a more involved issue, however.
August 9, 2012 at 3:37 pm
Whoah, 1337 h4x0r alert!!!!!!!