The name of the currency is euro, not capitalized. It is divided into 100
cent. Although the name of the subunit is cent (not eurocent), the coins actually say EURO CENT (one word above the other).
The official plurals in legislation are the same as the singular, euro and cent, at least in most community languages (including English). French and Spanish get to use plurals, and Finnish gets to use its partitive case as with other numerals. Oddly, the Greek for cent is lepto (the old 100th part of a drachma).
Coins will be issued for cent multiples and lower euro multiples (1 and 2 euro), and banknotes thereafter up to remarkably high values (5 to 500 euro); these latter feature bridges in various architectural styles through the ages (not intended to be depictions of specific places), and a map of Europe. Worthy but dull.
A little-mentioned fact (am I the only one to notice this?) is that the EMU (European Monetary Union) currencies were formerly lined in flexible but related exchange rates in the so-called snake. And a euro is a kind of wallaby. Why are our monetary masters naming everything after Australian fauna?