Iceland is a small island state (about 320,000 inhabitants) just south of the Arctic circle, in the North Atlantic between Europe and North America.
The capital region, where the vast majority of the population lives. The north with a minor urban center around Akureyri, and popular lake Mývatn. The south with many impressive waterfalls, volcanoes and glaciers. The often neglected east with fjords, mountains and fishing villages. The remote Westfjords with very few people, very bad roads, and lots of scenic landscape. The uninhabited highlands that make up the biggest part of the country.
Reykjavik (120,000) and some less important ones in the capital region. Akureyri (18,000) in the north, Egilsstaðir (2,300) in the east, Ísafjörður (4,000) in the Westfjords.
By plane: with Icelandair or http://www.wowiceland.co.uk/. There's also a number of foreign airlines serving Reykjavik/Keflavik and sometimes Akureyri, especially during summer. By boat: There's a ferry connection via the Faroe Islands to Hanstolm and Esbjerg in Denmark.
By infrequent and expensive overland buses, rental cars or domestic flights. For the more adventurous, Iceland is a great country for hitch-hiking, though you should be prepared for cold, rainy/snowy and stormy weather, and you may have to be patient at times.
What to do
Once you leave the randomly built and very car-centered urban areas, you can hardly do anything wrong - you'll find exciting and beautiful places everywhere.
Icelanders are generally friendly, funny, open-minded, and speak English fairly to very well. Oh yeah, and there's Björk and so on, but Icelanders don't make a fuss about celebrities - partly because they're so few that a large percentage of the population is somewhat famous or has relatives or friends that are, and a lot of people are involved in arts and culture.