It's impossible to overstate how much the scholastic landscape of early Medieval Europe was shaped by the machinations the Irish. At points basically every learned person on the continent was Irish. This results in things like Frankish courtiers essentially boycotting the King's service until he employs some native Franks rather than relying on the Irish, and Italian synods coining the term "Irish porridge" to describe weird, esoteric theology.
Ireland was also incredibly important in the Viking Age. The continued military victories of the Irish against the Vikings made sure that their raids were a lot less devestating to other parts of Britain and Europe than they could have been.
Read
Paolo Delogu, An introduction to medieval history (London, 2002)
Katherine Simms, Medieval Gaelic sources (Dublin, 2009)
Kathleen Hughes, Early Christian Ireland. Introduction to the sources (Cambridge, 1972)
Daniel Mc Carthy, The Irish Annals: Their genesis, evolution and history (Dublin, 2008)
Brian Tierney & Sidney Painter, Western Europe in the middle ages: 300–1475 (New York, 1999)
Henry Mayr-Harting, The coming of Christianity to Anglo-Saxon England (Oxford, 1972)
Francis J. Byrne, Irish kings and high-kings (Dublin, 2001)
Donnchadh Ó Corráin, ‘Ireland, Wales, Man, and the Hebrides’, in Peter Sawyer, The Oxford illustrated history of the Vikings (Oxford, 1997)
Francis John Byrne, ‘The Viking Age’, in Dáibhí Ó Cróinín, A new history of Ireland I: prehistoric and early Ireland (Oxford, 2005)
Peter H. Sawyer, Kings and Vikings: Scandinavia and Europe AD 700–1100 (London, 1982)
Francis J. Byrne, ‘Ireland and her neighbours, c.1014–c.1072’, in Dáibhí Ó Cróinín, A new history of Ireland I: prehistoric and early Ireland (Oxford, 2005)
Marie Therese Flanagan, The transformation of the Irish Church in the twelfth century (Woodbridge, 2010)