There were several kingdoms in Ireland which shifted territory over time. The 5 traditional kingdoms were ulster, leinster, connaught, munster and meath and monarchy was not a strictly herditary...
The most well known King of Ireland was The High King Brian Boru. A common name in Ireland is O'Brien, meaning "from brian". The name O'Brien indicates that the family are decendants of the High King...
This is not known for certain. One opinion is that two brothers, Heber and Heremon jointly ruled some Irish territory about 3700 years ago, and that they were the first Irish kings.