64 bit computers process data in larger chunks than 32 bit computers. This means a number of things, but the big 2 are:
1) When programs are written to support 64-bit processors, those programs will generally perform faster than on a 32-bit processor of the same speed. This is partially because 32-bit computers must handle 64-bit numbers in two steps, whereas 64-bit computers only need to use one step.
2) 64-bit processors can address more memory than 32-bit processors. This means your server can have more RAM, and therefore can store more data in memory, making memory-intensive programs like databases faster.
As for databases, yes, there are versions of several major databases out there that support 64-bit processors. The one that I am most familiar with is SQL Server of course. SQL Server 2000 is available in a 64-bit version, and is intended to run on the 64-bit version of Windows 2003 Server. Unfortunately, without the underlying 64-bit server OS, I dont believe the 64-bit version of SQL Server will give you any benefit (if it even runs).