The short answer: no, you cannot use Euros in Iceland.
Iceland, to the surprise of some, is not in the European Union, nor does it use the Euro. In fact, Iceland is the second-smallest nation (after the Seychelles) to maintain its own currency and monetary policy, which is called the Icelandic króna, or ISK. It is, however, a part of the Schengen zone, which allows freedom of movement for citizens of partner nations.
Some of Iceland’s Nordic peers, do, however, have different arrangements. For instance, Denmark is part of the EU, but also retains its own currency, the Danish krone, which is pegged in value to the Euro. All of the Scandinavian nations have their own currencies, which are all variations on the local word for “crown,” just like there is a US, Canadian, and Australian dollar. These Scandinavian currencies are descended from the Scandinavian Monetary Union, a monetary union between Denmark, Norway, and Sweden in the 19th century. It stemmed out of a growing desire among some to unite these similar nations in a “pan-Scandinavian” movement. Little else came of this, but the monetary union was generally considered successful and helped to tie the region’s economies together under a standardized system.
But these days, the Icelandic króna is the only official currency, so plan accordingly.
If you are planning a trip to Iceland, know that digital transactions are largely the rule in most establishments. You certainly can pay in cash, but most places of business prefer electronic payment for convenience. Still, a well-prepared traveller should exchange some cash to have on hand, just in case. It is possible to change cash at the airport or at a bank in town. You can also use an ATM (in Icelandic: Hraðbanki) to withdraw local currency, but these generally come with fees.
However, some large stores, attractions, and restaurants that deal largely with tourists may accept Euros, USD, or other major currencies. Also, because of the high volume of connecting flights through Keflavík International Airport, all merchants at the airport accept Euro, USD, and some other major currencies. It is worth noting that this is not an official policy: it is a service that some businesses provide their customers with for convenience. That means that while it may be an option, do not make your plans around being able to pay in Euro, USD, or other currencies.