Founding of the Republic

A Vote of Consequence 

On 31 December 1943, the Danish–Icelandic Act of Union expired after 25 years.
Beginning on 20 May 1944. The Icelandic people then voted in a four-day election on whether or not to terminate the personal union with Denmark, abolish the monarchy, and establish a republic.
The vote was 97% in favor of ending the union and 95% in favor of the new republican constitution.
The Icelandic people wanted to form a republic, so they did.

Formal Ceremony

The establishment of the republic of Iceland formally took place at Þingvellir (Thingvellir) on June 17 1944.
At 13:30, the prime minister of Iceland, Björn Þórðarson, officially set the celebrations, after which a Christian mass was held.
The new flag of the Republic of Iceland was raised, and the members of the Icelandic parliament rose from their seats, as church bells rang.
All declared unilaterally that Iceland would henceforth be a republic. The members of the Icelandic parliament then voted on who should be the first President of the Republic, and chose Sveinn Björnsson, who had been regent of Iceland and the King's placeholder during the war years.
Sveinn thus became the first president of Iceland, and the only one not elected directly by the people of Iceland.

 

The Republic of Iceland today

Iceland is a representative democracy and a parliamentary republic.
The modern parliament of Iceland, Alþingi (English: Althing), was founded in 1845 as an advisory body to the Danish monarch.
It was widely seen as a re-establishment of the assembly founded in 930 during the Commonwealth period and suspended in 1799. Consequently, it is arguably the world's oldest parliamentary democracy
It currently has 63 members, elected for a maximum period of four years. 
The president of Iceland is elected by popular vote for a term of four years, with no term limit.
The elections for president, the Althing and local municipal councils are all held separately every four years.