An EORI (Economic Operators Registration and Identification) number is a unique identifier required by businesses that import or export goods into or out of the EU. In the UK for example, it allows HMRC to identify which business has paid the import VAT on goods, which is required for reclaiming on your VAT return.
You can obtain your EORI number as part of your VAT registration. If you do not get one, you may have increased costs and delays. For example, if HMRC cannot clear your goods you may have to pay storage fees. Visit HMRC's guidance on EORI page for further detail.