The Irish, Welsh, and Scots all have a Celtic ancestry, but they settled their respective regions before the Roman conquest of Britain.  There is an amazing amount of debate as to the origin of the Celts:  were they Phoenician?  stocky and dark?  tall and blonde?  as culturally cohesive as the label suggests?   The standard theory is that the Celts were an Indo-European group that gradually migrated across Europe and Asia, with an identifiable, distinct culture by 750 BC.  As a group, they were well-known to the Greeks and Romans.

http://archaeology.suite101.com/article.cfm/archaeology_and_the_celts

The Celts had arrived in Britain and Ireland by 400 BC, super-imposing upon whatever native peoples were already there.  The Celts in these regions, then, were on the fringes of Celtic culture, not their heart, which was centered in Northern Europe, particularly in what is now Germany, Austria, and Switzerland.

http://www.knowth.com/the-celts.htm

The links between Wales and Ireland continued to hold through the Roman conquest and the years after.  There is strong evidence of a continued Irish presence in Wales, particularly on the west coast of Wales.  The rulers of Dyfed were of Irish descent into the 7th century–and there is also evidence of repeated raids from Ireland to Wales.

According to Thomas:

“… both Irish and Welsh sources portrayed it as a tribal migration of the Irish Dessi or Deisi headed by their own king and, from the Irish viewpoint, a suitable ‘expulsion’ saga was adduced. The direct line of Irish rulers of Welsh Dyfed went on into the 7th and 8th centuries. An interesting mix arose; by 400 Irish and British were fully differing languages, and additionally Christians from both nations used different scripts (Latin and Ogham) for their memorials. Irish never replaced British in Wales the way it did in Scotland, but relative numerical strengths do not necessarily explain why; less obvious factors could be involved.”

http://www.islandguide.co.uk/history/ogham.htm

Within Welsh mythology, the Irish play a significant role as well.  Taliesin sings of himself:  I have been with Bran in Ireland.  This is in reference to the tale of Bran the Blessed who obtains a magical cauldron from Cerridwen (in disguise as a giantess).  She had been expelled from a lake in Ireland. The cauldron can resurrect the corpse of dead warriors placed inside it (this scene is believed to be depicted on the  Gundestrup cauldron):

http://www.unc.edu/celtic/catalogue/Gundestrup/kauldron.html.

Bran gives his sister Branwen and her new husband Matholwch — the King of Ireland, and not to be confused with Math ap Mathonwy, the King of Gwynedd — the cauldron as a wedding gift, but when war breaks out Bran sets out to take the valuable gift back. He is accompanied by a band of a loyal knights with him, but only seven return home.   A similar tale is told in Taliesin’s poem, the Spoils of Annwn about King Arthur’s descent to the Underworld.

In the Middle Ages, there was much back and forth between the rulers of Wales and the rulers of Ireland.   Not only did they share ancestry and blood, but retreated one to the other at various times when they were driven out of their own kingdom (in the case of Gwynedd, due to usurpers or the Normans). In one specific case, Owain Gwynedd’s father, Gruffydd ap Cynan, claimed ancestry to both the Norse kingdom of Dublin and to the Celtic High Kings of Ireland:

“According to the Life of Gruffudd ap Cynan, Gruffudd was born in Dublin and reared near Swords, County Dublin in Ireland. He was the son of a Welsh Prince, Cynan ap Iago, who was a claimant to the Kingship of Gwynedd but was probably never king of Gwynedd, though his father, Gruffudd’s grandfather, Iago ab Idwal ap Meurig had ruled Gwynedd from 1023 to 1039. When Gruffudd first appeared on the scene in Wales the Welsh annals several times refer to him as “grandson of Iago” rather than the more usual “son of Cynan”, indicating that his father was little known in Wales. Cynan ap Iago seems to have died while Gruffudd was still young, since the History describes his mother telling him who his father was.

Gruffudd’s mother Ragnhild was the daughter of Olaf of Dublin, son of King Sigtrygg Silkbeard and a member of the Hiberno-Norse Uí Ímhair dynasty.[1] Through his mother, who appears in the list of the fair women of Ireland in the Book of Leinster, Gruffudd claimed relationships with many of the leading septs in Ireland. His great-great grandparents on his mother’s side include the High King of Ireland, Brian Bóruma, and the King of Dublin and King of Northumbria, Olaf Cuarán, and Gormflaith.[1]

During his many struggles to gain the kingship of Gwynedd, Gruffudd received considerable aid from Ireland, both from the Hiberno-Norse at Dublin, but also those at Wexford, and also from Muircheartach Ua Briain.”  http://en.wikipedia.org/wiki/Gruffudd_ap_Cynan