Well, you can create custom labels for existing field types, but that probably isn't what you want.
There are probably other address apps that will do what you want, but the only one I'm familiar with is Bento.
Bento isn't actually an address book. It's a database, but it automatically imports and syncs with the existing address book on the iPad. The benefit is that you can add custom fields to the Bento version. They won't export back to the address book, but any changes you make to the normal fields will.
If you happen to be using a Mac you can get the OS X version of Bento as well. This gives you considerably more options in designing and integrating your address book with other databases created in Bento.
I don't recommend Bento on the iPad as a stand alone database for anything but small flat projects, but it does make a nice address book, and it makes a good personal data base if you've got the OS X version to sync with.