All you need is your laptop's serial number plus your battery's serial number for the first page. On the second page, your personal information such as the address you want the battery to be shipped to.
Word of warning. Make sure you confirm your address is correct before you submit the second page. There's no confirmation page that will say "Are you sure this is correct? Hit back if you are not." If you screw up, you'll probably have to contact Apple again to correct your address. Either that, or camp out at your neighbor's house daily for 4-6 weeks.
Battery Exchange [Apple]