According to Iceland's 100% literate population, you can't go past this book by Ari Þorgilsson (known respectfully as Ari the Learned) for an authoritative account of the country's human history. Don't let the fact that it was written in their native tongue in the 12th century deter you from reading it.
Iceland Saga
by Magnús Magnússon
For an easily digestible account of Iceland's past and its historicised folklore, check out this book.
The History of Icelandic Literature
by Stefán Einarsson
Gives a thorough overview of the nation's scribblings.
Journey to the Centre of the Earth
by Jules Verne
This classic tale of adventure includes some descriptions of Reykjavík.
Moon Country - Further Reports from Iceland
by Simon Armitage & Glyn Maxwell
Entertainingly describes an attempt by modern-day poets to retrace the icy footsteps of WH Auden and Louis MacNeice, who write about their own irreverent journey in 1937's Letters from Iceland.