Eurobonds are named after the currency they are denominated in. For example, Euroyen and Eurodollar bonds are denominated in Japanese yen and American dollars respectively. A Eurobond is normally a bearer bond, payable to the bearer. It is also free of withholding tax. The bank will pay the holder of the coupon the interest payment due. Usually, no official records are kept.
The majority of Eurobonds are now owned in 'electronic' rather than physical form. The bonds are held and traded within one of the clearing systems (Euroclear and Clearstream being the most common). Coupons are paid electronically via the clearing systems to the holder of the Eurobond (or their nominee account).