EORI stands for Economic Operator Registration and Identification and is a registration number for anyone, whether VAT registered or not, who is invloved with importing and/or exporting. The legislation restricts the issue of a single EORI number to a legal entity - sole proprietor, partnership, company or individual. It therefore cannot be allocated to individual branches or divisions within a legal entity.
Basically it is a security measure. Once registered your details are kept on the National and EU database.
UK EORI numbers are based on the format of the expired Traders Unique Reference Numbers (TURNs). They start with the letters 'GB' and then most are followed by a 12-digit number based on the traders VAT number (eg GB123456789000).
In general HMRC with notify you or your EORI number within 72 hours of application.
If you are VAT registered click here
If you are not VAT registered click here
Further information can be found on the HMRC website