There are both 32-bit and 64-bit versions of Windows 7 Home Premium, although most new Desktop PCs will probably have the 64-bit version installed. 32-bit programs should run without problems in 64-bit Windows, but peripherals such as printers, scanners etc will require the 64-bit drivers.
A router is a device that allows multiple computers to access the internet simultaneously. However, as versions of Windows after XP (i.e. Vista and 7) don't generally have suitable drivers for USB ADSL Modems, ISPs moved over to providing combined modem/routers for internet access even for a single PC.
With a modem/router, it is that which connects to the internet, not the PC which gets its internet access over the network by connecting to the router by ethernet cable or "wirelessly".