The birth of the euro has raised hopes by some, and fears by others, that it might, sooner or later, challenge the U.S. dollar's position as the dominant international currency.
Any attempt at addressing this issue can only be tentative. Indeed, a currency's international status is not granted by official fiat but results from concurrent choices made by public and private operators world-wide.
Many currencies quite naturally play an international role as the invoicing and payment unit of their countries' foreign trade. Among the industrialised countries, the U.S. succeeds best in doing so. But most of the world's 200 or so countries cannot impose their own currencies in payment for their imports and may even want their exports to be paid, not in their own currencies, but in a more reliable or usable foreign currency, such as the U.S. dollar.
This is, in fact, what is meant by the "international" role of the dollar: its use between third countries in a way similar to its use within the U.S.
Indeed, the dollar, which at the domestic level is used as a unit of account, means of payment and store of value, performs the same functions between countries for both their private and public sectors :

•  the private sector: invoicing and payment in dollars of their export/import of goods and services with countries other than the U.S.; holdings of dollar assets such as deposits, bonds, etc.;
•  the public sector: dollar "anchor" for their currencies' exchange rate (fixed-, or sliding peg); choice of the dollar as the intervention currency on their exchange markets; dollar as main reserve currency of their central banks.
Moreover, beyond its use between countries, the dollar may also be used within a given country. The most popular form of this so-called "dollarisation" is the parallel use of dollar banknotes in some emerging economies. Its most extreme form - under consideration in Argentina - is the adoption of the dollar as the national currency.