The first settler of Iceland, Ingolfur Arnarson, is believed to have settled in Reykjavik in the year 870 ad. He had started his journey along with his step-brother and decided to seek the land that Hrafna-Flóki had described and called Iceland.
When Ingolfur saw land, he threw his pillars overboard and pledged to settle where they would find land. He stopped where now is known to be Ingolfshofdi but Hjorleifur, his step-brother, at Hjorleifshofdi.
At Hjorleifshofdi, the slaves of Hjorleifur went at him and killed him and his people. They fled and found land at an Island of the south coast. When Ingolfur found his brother dead, he became furious and followed the slaves and found them in the Island. He killed them all. Since, the Islands have been known as the Westman Islands.
Ingolfur sent two of his slaves, Karli and Vifli to find his pillars. It took them three years. Ingolfur went there and built his farm. It is believed that the farm is where now is Adalstraeti in the center of Reykjavik. There is a museum there where you can see the ruins of the settlement farm and read everything about the settlers.
See the website of the Settlement exhibition