In ancient Ireland the population was much smaller than today and the mass movement of people was uncommon. It was usual therefore for a person to be known only by one name: Niall, Eoin, Art, etc. Once there was no one else in the locality with the same name then this was not a problem.
The Gaelic Clann system was well established and this gave people a common identity with their people of the tribe and with the commonly shared area. This single name system began to break down during the eleventh century as the population was growing and there was a need for a further means of identification. The solution was to adopt a prefix such as Mac (Mc is an abreviation) or Ó. Mac means 'son of' whilst Ó means 'grandson of'. Mac surnames are generally of a much later date than Ó. The vast majority of Gaelic Irish surnames were created during the eleventh and twelfth centuries.
Septs: The Clans eventually broke up into a number of distinct septs or groups. These groups were headed by an original member of the clan and dominated a particular part of the countryside. It was not uncommon for septs from the same clan to be found in completely different parts of the country (O'Connor for example) so it is important when researching your roots to try to find out the original part of the country that your ancestors came from as this may be a completely different area from that where the 'major' sept was domicile.
The sept system was an integral part of Gaelic society and survived and was even propagated by the Norman invaders. The system did not survive the English invasion and colonisation of the seventeenth century however, and it became a disadvantage to have a Gaelic sounding name.