A VAT number (or value added tax identification number) is a number used by countries within EU (and outside) for purposes related to VAT or Value Added Tax.
On the Record Union website, a VAT number is needed in order to receive royalty payments as a company based in the EU. Without this number we would not be able to comply with the EU regulations governing sending payments to other companies.
Not sure what your VAT number is?
VAT numbers may have a different name in the country you are located in.
Please see this Wikipedia article which provides an additional country specific info concerning VAT numbers.