Depending on how long you intend to stay in Iceland there are various options. In almost all cases there are more houses/apartments for sale than for rent, and in many cases it might take some time to find an apartment which suits your specific needs. However almost all websites are purely in Icelandic you will have to rely on some sort of online translation in beginning.
For short term – low budget
http://www.hostel.is/
For short term – high budget
http://www.rent.is – Apartments for short term rent that are fully furnished
Long term rent
Rent for an medium sized apartment in the Greater Reykjavik area runs from 70.000 – 150.000 ISK and above, at current exchange rates in June 2012 that is equivilant to 420 – 900 Euro.
Currently all the websites for long term rent are in Icelandic, you can use online translators such as Google translate – http://translate.google.com/ to translate single words or the whole webpage. Be aware of translation errors!
mbl.is Free apartment search – Only in Icelandic
visir.is Free apartment search – Only in Icelandic
Leigulistinn.is – One of the most popular rent markets, subscribtion access to the website for one month costs 3.600 ISK (approx. 20 euro)
In most cases a valid Icelandic social security number, and a Icelandic bank account are needed to sign a lease. Also in most cases you are required to pay up to 3 months rent as insurance.