Modern processors are all 64 bit. Modern OSs are available in 32 bit and 64 bit variants (Windows XP is not a modern OS, but it is very popular, so Microsoft released a 64 bit variant).
64 bit processors are perfectly compatible with 32 bit operating systems, but 64 bit operating systems are not backwards compatible with 32 bit applications per se, although most 32 bit applications can be run on 64 bit systems with some tricks.
On a desktop computer, the main reason for having a 64 bit OS is the RAM. A 32 bit OS cannot use more than ~3.5 GB memory (here "memory" is the sum of system RAM, graphic card RAM and some firmware chips on the mainboard and the extension boards). Computers sold nowadays usually have more than 3 GB RAM, so for them, a 64 bit OS isn't a luxury, it is a necessity. And in the future, the share of 64 bit applications will grow and the share of 32 bit will shrink, so if you are buying a computer right now, there is no sense in getting a 32 bit OS. But if you already have a 32 bit OS, there is no need to change to 64 bit, unless you have some very good reason to upgrade your RAM.
As for how to check which you have, just go to Control Panel and open System. It will tell you whether your current OS is 32 bit or 64 bit. If you don't want to upgrade, use the knowledge to decide which program versions to download and keep it at that. If you plan to upgrade, you'll see your processor type listed there, for example, "Intel Core2 Duo T9300". You can then look on the Internet whether the processor supports 64 bit or not.
Related: http://superuser.com/questions/56540/32-bit-vs-64-bit-systems
– BloodPhilia – 2011-02-13T12:59:16.963