Irish nobility falls into three categories:
Gaelic nobility - those who descend from the kings of Ireland
Peerage of Ireland - those who descend from ancestors given titles by the British Monarchy
These groups overlap with a high degree, particularly the Gaelic and Hiberno-Norman or Old English nobility.
There where also (and still are) Chiefs of the Name. The position of Chief of the Name was more important to some Irish leaders than English titles. The Chief of the Name can be described as 'the head of the clan'. Traced through the male line the title was either passed down via primogeniture or derbfine.