32 bit vs 64 bit refers to the address space, and as others mentioned this translates into the amount of RAM you can use: 32 bit = 4GB, 64bit has a theoretical ceiling of 16, 384 petabytes. Wow.
The fact that you are posting on superuser probably indicates that you aren't a "normal" user. If all you're doing is checking email and writing word docs, you are probably ok with 32-bit; however, if you are buying a new computer and you're not looking for a cheap netbook, I see no real valid reason NOT to go with a 64-bit system. There are, however, many reason to get a 64-bit and thus as much RAM as you can get your grubby hands on, including : compilation performance if you're a programmer, multitasking performance, heavy duty video/graphical editing performance, VM performance, and high-end gaming performance. Notice the keyword performance? If that is important to you in the future, go for 64-bit.
propose closing this as a duplicate of http://superuser.com/questions/56540/32-bit-vs-64-bit-systems
– quack quixote – 2010-02-27T05:38:24.847