The Economic Operator Registration and Identification (EORI) scheme started on 1 July 2009 and replaced the previous TURN system. The EORI provides a unique number for a business to quote to any Customs authority in the EU, for example when the business needs to make a customs declaration when goods arrive or depart from the EU.
If your business had a number under TURN, it should automatically have been issued with an EORI number.
If your business is VAT registered in the UK you will need to complete form C220 to obtain an EORI number, if you are not VAT registered the form to complete is C220A.