This question pops up quite a lot in online forums nowadays. Before I could quite honestly tell somebody if you need to ask, you have 32-bit. That's not true anymore, you can buy machines with 4GB of RAM in them, and increasingly they are being loaded with 64-bit Windows Vista, so the system can make use of all the memory.
The following works for Windows Vista and Windows 7:
You can find out by going to Control Panel -> System and Maintenance (System and Security for Windows 7 users) -> System.
Alternatively you can search for System on Start Search, the search will return a few results, but one should just be the word System, with a computer icon next to it with a white tick on a blue monitor. That's the one you're after.
Clicking on that will launch the following.
System type displays if the system is 32-bit (often referred to as x86) or 64-bit (also known as x64 or x86-64).
For Windows XP:
If you're running Home Edition, you're using 32-bit. Professional shipped as 32-bit and 64-bit, but almost nobody uses x64. To check go to Start -> Right-click on My Computer -> Select Properties from the Menu, from there the 64-bit versions will be listed as Microsoft Windows XP Professional x64 Edition. Otherwise you're running 32-bit.