In 1995 when Windows 95 was released it was introduced the x32 architecture. Back in the days this architecture was requested because it could handle large amounts of RAM(4 Gigabytes) for that time. Today modern average PCs has more than 4 GB of RAM which is actually a minimum required, but back in 95 it was a huge technological progress.
In 1999 the x64 architecture was introduced because the old one couldn't not handle anymore the technological progress, there was already users with files(games or videos) larger than 4GB which cannot run on old system.
A x32 bit operating system will run on a x32 or x64 architecture processor.
A x32 bit application can run on a x32 or x64 operating system.
A x64 bit application will run only on a x64 bit operating system and a x64 bit operating system will work only on a x64bit processor.
Today is recommended to have a x64 processor with a x64 operating system installed.
More bits are much better, more bits means more data can be processed which is actually improving computer performance.