Well...a 32-bit system can only employ up to 4GB of RAM, whereas a 64-bit system hypothetically can employ up to 128 GB (I heard rumors of more).
I think in terms of a MOBO this would mean that there are either 2 or 4 channels on the MOBO. with a 32-bit system you can have 4-1gb sticks, running in dual channel (two sets). Or 2-2gb sticks, if there are only two channels.
If it is a 64-bit system, the MOBO manufacturer is implying that the MOBO is only rated for 4-2bg sticks (most likely).
Well operating systems are either based off of 32-bit or 64-bit architecture, which defines how they handle information - the number of bits in each pass (essentially). Windows XP, Vista, and Windows 7 all have 32 and 64-bit versions. The bit-size is defined by the OS manufacturer - and then there is essentially two versions of the OS, one in 32-bit and one in 64-bit. The Windows 7 DVD will have both 32 and 64-bit versions on the same DVD, and you get to choose which you want to install.
I am not sure how it worked for Windows Vista, but with XP you had to buy a separate installation CD for the 64-bit version, and the default (masses) version was 32-bit.
I am not familiar with Debian, but here is some information regarding the 64-bit version. So it appears that there is both a 32-bit and 64-bit version of Debian.
It means that a 32Bit Operating System supports up to 4Gb and a 64Bit Operating System Supports 8GB or More!
There is no difference between the ram,There is only difference between the amound of ram and the bits!