The later high kings were nominally in charge but in practice, descended as many were from the prominent U Neill clan, there were always stresses and strains with the other regions.
For the earlier high kings, the title was more of a ceremonial one, and never implied political control of the whole country.
Ierne is fairly obviously a mispronunciation of 'Er Inis' or 'Eire Innis' (various spellings are available), meaning 'West Island' in common Celtic. The name remains in use today in its full form - Eireann.
(Additional information by Edward Dawson, and from Marie Mc Keown, Hub Pages.) Until now a land bridge has connected Britain to Ireland, roughly from the south-eastern tip of the latter to south-western England.
At some point after about 500 BC, there were certainly arrivals by Indo-European Celts (and perhaps even as early as 2000 BC), and they remained fully independent as Ireland was never conquered by the Romans.