We recently visited Reykjavik, Iceland and were pleasantly surprised by the VERY high quality of the restaurants we dined at.
Our most memorable was the GALLERY RESTAURANT AT HOTEL HOLT. This is acknowledged as the best restaurant in the country - and we certainly were not disappointed. Yes, it's located in a hotel but please don't let that stop you. The owner of the hotel has Iceland's largest private collection of art which is prominently displayed in the restaurant - hence the name. We had Icelandic lamb there which was the best lamb I've ever tasted. Honestly. The service is exemplary. It's pricey like everything in the country - but worth it.
Another restaurant which is supposed to be in the same league is the PERLAN. It is a revolving restaurant and heavily advertised to tourists. We were told by people who'd been there that it was not a good value: filled with tourists and factory-like service. You can visit the viewing platform at the restaurant without dining there.
A very cozy restaurant that we liked was called THRIR FRAKKAR - which means Three Frenchmen in Icelandic. It's small and very authentic. The quality of the food is superb. The service was a pit patchy, however. They did not have enough staff on hand. It's mainly at seafood restaurant although meats are also on the menu. (Despite its name, its not a French restaurant.)
We also went to the APOTEK restaurant. This is a very modern and trendy place, which had been recommended to us. However, we found the food a bit too modern and trendy for our taste. The menu is Icelandic-Japanese "fusion" for the most part. Perhaps others will enjoy it more.
A great Italian restaurant we liked was LA PRIMAVERA. It was packed with locals and Italians, which was a very good sign! We had pasta but they also serve Icelandic cuisine with an Italian accent.
There a very useful website that gives a listing of restaurants, and provides links to their own webpages where many of them display they're menus in English.