In ancient Ireland the population was much smaller and the mass movement of people was uncommon; therefore, it was usual for a person to be known only by one name.  The Gaelic Clann system was well established and this gave people a common identity with their people.  This single name system began to break down during the  eleventh century as the population was growing.  The solution was to adopt a prefix such as Mac (Mc is an abbreviation) or O.  Mac means 'son of' whilst O means 'grandson of'. 
 Mac surnames are generally of a much later date that O.  
Note:  The Scottish Gaels were actually descendants of Gaelic emigrants to Scotland.  The word 'Scotus' is Latin for 'Irishman'.  Scottish settlers who moved to Ireland (and especially Ulster) may already have been of Gaelic Irish descent.