The 32 bit and 64 bit refer to the size a memory address (addressing a single byte) has. A greatest 32-bit number is 4,294,967,295, which means that 4 GB can be addressed. So a 32-bit OS can only use 4 GB of memory, tricks and tweaks aside. In these 4 GB of so-called "address space" however, there's not just your RAM. There are also reserved memory regions for almost every internal device, including your graphics card. Which obviously, depending on the amount of graphics memory, blocks shitloads of the 4 GB address space from being used to access RAM. As you can see, a 32-bit OS is severely limited in terms of usable RAM, by today's standards.
A 64-bit OS on the other hand, can address (in theory, today's systems can't) the incredible amount of 18,446,744,073,709,551,616 bytes. That's 16 exabyte. No problem's of any sort there, at least for the next couple of years.
Bottom line is: If you have 4 GB RAM or more, you need to use a 64-bit OS, or you won't be able to take advantage of all that RAM.