When Iceland was permanently settled from 874 AD, the settlers from Norway brought with them wives and thralls (þrælar – slaves/serfs) from Ireland. The Norwegians spoke a form of Old Norse, which developed into modern Icelandic, and borrowed a few words from the Irish.

Here are some examples:

• á = river – from the Old Irish aub (river). The Old Norse word for river was fljót.
• áin = (act of) driving – from Old Irish áin = (act of driving (animals etc))
• bagall = crozier, from the Old Irish bachall (staff, crook, crozier). The Old Norse word for crozier / staff was stafr.
• brekán = blanket, quilt, from the Old Irish breccán (speckled thing, striped or chequered stuff, plaid)
• bjannak = blessing, from the Old Irish bennacht (blessing)
• tarfur = bull, from the Old Irish tarb (bull). The Old Norse word for bull was þjórr.

Icelandic has also absorbed words from Latin, Greek, Middle Low German and Danish. Some of these words are no longer used, but new loanwords, mainly from Danish and English, are streaming into the language, especially in the fields of IT and medicine. One example I heard about was móment being used instead of augnablik (‘eye-blink’) for moment.

Sources: Wiktionary, Germanic.eu, eDil