Hotels that have a formal classification from the Icelandic Tourist board are specially marked with their shield of approval. Here below is an exerpt from their website´s classification text:
(Quote begins:)

Classification of accommodation in Iceland

"All lodging facilities in Iceland have the oppurtunity to be classified. The classification is divided into five categories i.e. 1,2,3,4 or 5 stars and is by no means an all-embracing assessment of the quality of the accommodation in question. The grading consists of more than 100 different items and is solely based on factors which can be objectively measured for exemple whether specific facilities or services are available or not. The Icelandic Tourist Board is in charge of the classification system. Below is a brief listing within each catergory:

One star: (See P.s. on bottom of page)
All rooms are equipped with washbasin and there is at least one bathroom for every 10 rooms. There is access to a public telephone, breakfast is available and guests can come and go any time of the day or night.

Two stars: 
In addition to the one star facilities, there is a telephone booth or a public telephone where the guests can speak in privacy. There is a bar or service counter where it is possible to buy light refreshments, as well as a dining room where breakfast is served, and other meals, depending on circumstances. The rooms may be equipped with a private bathroom, but this is not a condition.
  
Three stars:
In addition to the class above, all rooms are equipped with a private bathroom, telephone, televison, radio and desk. There is a service counter open during the day and it is possible to buy small items, reading materials etc. Goods and services can be paid for by credit card. There is a lift in the building, if guest rooms are three or more levels above the reception.
 
Four stars:
In addition there are easy chairs in all rooms. TV with remote control, satellite channels and a movie or video channel. There is room service day and night, or a minibar, an “a la carte” restaurant, and breakfast can be ordered in guest rooms. There is also a clothes cleaning service. There is a lift in the building, if the guest rooms are on two or more levels above the reception."

(Quote ends: )

More information is available at the website of the Icelandic Tourist board: http://www.ferdamalastofa.is/flokkun/en.htm

P.s. There are no 5 star or 1 star hotels in Iceland. No 5 star because they don´t fulfill all requirements. Often things such as a swimmingpool is missing but many of the 4 star hotels are 5 stars in service and feeling. There are no 1 star hotels., I think, because such a hotel doesn´t want to pay the fees needed for a formal 1 star rating. If you think of it, why pay somebody to verify that you are a 1 star hotel??