right click on 'my computer', highlight 'properties', and click. One of the tabs tells you about the operating system, version and bit, along with your installed RAM.
software is generally compatible one version back. That is, 64bit will run 32bit software, but not 16bit. 32bit OS will run 16, but not 8.
If you bought your computer within the last year or two, it's probably a 64-bit processor, which will run 64 and 32 bit software.
If you bought your computer any time after they stopped booting to command lines (i.e. that black screen where you type in the commands), it's probably 32 bit. It'll run 32 bit software, and I guess 16 if you can find it?
Hardware designers will probably keep exploring multi-processor 64-bit systems for a while until we start hitting the upper limit of that hardware. (Something like a terrabyte of RAM, I think.) Then they'll switch up to 128-bits.
That, or quantum computing will obsolete the whole shebang.