Up to about 1170 Ireland was a largely disunified arrangement of kingdoms although there were High Kings, the island was never really a united entity - petty kings ruled various areas (They would nowadays be called 'war-lords'.)
Into this came the English at the invitation of one of the powerful who wished for assistance in regaining his kingdom.... and they stayed for a bit longer than they were welcome.
"The English" later became "The British" as England merged with Scotland.
They ruled Irish before, during and after the great famine of the 1840s.