As a native Icelander, I've been asked several times why there are so many cafés in Iceland. Since I am not the wizard of Reykjavík, I do not have the answer. But I do have a theory. In the winter, the weather doesn't often allow a whole day outside. At some point, you're going have to go inside and pour some hot liquid into your system in order to carry on with your day. Then, come summer, the days don't only get warmer, they get longer. Much longer. As the sun stays up the whole day and most of the night, you're going to want to do so too. What do you do in order to stay up? You go to a café and pour a drink full of caffeine into your system.
To really experience the city you must do as the locals do—or, as they say: When in Reykjavík, do as the vikings do. No, don't go around stealing all the women...the modern, civilized ReykjaViking drinks coffee and reads a book at his favorite café. Now the only thing you have to do is pick your favorite.
My all time favorite café in central Reykjavík is Mokka Kaffi (Mocha Coffee...obviously). It is quiet, full of regulars that come every day and old people who've been going there since they were in high school. The coffee is good but their speciality—the Belgian waffles — are the best! Every month, a new struggling artist puts up an art show there, so the walls are ever changing. In fact, they are the only things that ever change. Mokka Kaffi has kept the same appearance since it first opened in 1958.
For something a bit more fancy, Café Paris (no need for translation there) offers a full menu and is quite large, but it has the atmosphere of a small, cozy café. Their traditional Icelandic meat soup is very filling and should keep you warm and energetic for the rest of the day.
Stofan ("the living room") is a homey little place that probably looks more like your grandmother's living room than a café. They offer a variation of comfy, worn-out sofas, pillows, and one big dining table. They play nice music, and although their coffee is good, a cold beer from a local brewery is always tempting. The atmosphere is perfect for a game of Othello in a company of friends. Stofan is also a good place to study or work, with their free network and a power plug in every corner. And to top it all off, the bartenders there are almost always very friendly and, well, hot.
Those are my three favorite cafés in Reykjavík, but I encourage everyone to go try for themselves. Check a few places out, see how you feel there, and when you feel at home, you've found your spot.