To be frank, we don't really know that much. The historical period in Ireland begins only after the collapse of Roman authority in Britain so we have no contemporary records about this period. One particular problem about looking at Irish prehistory is that sources from the historical period do describe wars, kingly genealogies and so on from the prehistoric period, but these sources are imposing contemporary understandings of political order and relationships on the past. Essentially, early and high-medieval Irish historians were interpreting Irish prehistory in the terms of their own society, describing high-kings and systems of dynastic succession that just did not exist in the period that you're interested in.

So there probably weren't dynastic kingships and there certainly weren't high-kings during this period. However, there are some things that we can infer about Irish prehistoric society from later sources. Before the emergence of dynastic kingships during the fall of Roman Britain, Irish people were probably politically organized in a sort of theocratic system of cult allegiance, where extended kin relations were not very important. This seems likely based on the demonyms of archaic population groups mentioned in several earlier sources: unlike the later dynastic population groups named after a semi-mythical ancestor, these peoples apparently took their names from totemic animals. These demonyms end in the suffix -raige which means 'people of the -' which was combined with animal names to produce names like Artraige ('people of the bear'), Bibraige (people of the beaver) and Cattraige (people of the cat) and so on. Druids were just as important in pre-Christian Irish society as they were in Britain and the Continent so it isn't unlikely that they had significant political roles in this kind of theocratic society.

Ireland wasn't entirely culturally homogeneous during this period either. While Goidelic speakers were probably the most significant population on the island, there seem to have been other ethno-linguistic groups present as well. There were certainly Brythonic-speaking Celts present in Ulster, though by the historical period they were culturally assimilated by the Irish, only retaining their original demonym - Cruithin (the Irish word for Brythonic Celts). In the south, a historically significant people called the Erainn appear to have had non-Irish origins. An early medieval dictionary called Cormac's Glossary mentions that the Erainn spoke a language called the 'iron tongue' that was supposedly incredibly difficult to understand. It is possible that they were Belgic or Brythonic Celtic speakers that colonized southern Munster sometime in the the prehistoric period. As well, it seems that a non-Indo-European language was spoken in Ireland until the beginning of the historical period at the latest, which seems attested by linguistic evidence. Basically, there is a distinction between Q and P-Celtic languages, where all Goidelic languages used Q sounds in lieu of P or B ones. Sometime around the 5th century there was a shift and Irish began adopting words with those P and B sounds instead of altering them to Q ones. There are a number of loanwords of non-Indo-European origin in the Irish language that possess those P and B sounds, meaning that they were probably adopted pretty late in Irish history.