http://en.wikipedia.org/wiki/Windows_7_editions Wiki's chart is a bit more in depth. As said ultimate has more features but if you don't use them then it's a waste of money. The average person does not use them, they are mostly remoting and virtualization features.
Are you planning on joining your PC to a windows domain? If not then Home Edition will work just fine for you.
When you buy the Ultimate edition you not only get additional features but you also get both 32 bit and 64 bit versions of the OS in two seperate disks so if you get the Ultimate edition be sure to pay attention to which disk you are loading.
For most people home premium is more than enough. Ultimate isn't 'better' as in being faster or anything like that, it simply has a lot of additional features for advanced networking and user management type stuff for businesses.
The most notable advantage to Win7 Professional/Ultimate for an average user is WindowsXP mode; which will let you download a special version of windows XP that will run inside windows 7 (called a virtual machine), and you could run programs there that are not compatible with windows 7 (which are few and far between, usually old games and old business applications).
You get 32 bit and 64 bit with home and pro as well, just need the retail or upgrade version. This goes for ultimate too.
@shiny tech. Not that it matters. (or you'll be able to find it for sell anywhere) but there's a windows 7 home basic too. It supports 8gb .