Old English is a term with two meanings.
For the language Old English, see Old English language
Old English is a term used to describe a wave of early medieval English and Welsh settlers who went to Ireland to claim territory and lands. Many of the Old English came with the Norman invasion. Though English governments expected the Old English to promote English rule in Ireland, many soon abandoned their English identity, adopted the native Irish language and religious customs and marrying the mere Irish (contemporary term meaning pure Irish, mere meaning pure in contemporary english), and came to be called more Irish than the Irish themselves.
Ironically, England had only just been settled by Angles (and Saxons and Jutes) at this time-indeed some of the early settlers may have come over before the Anglo-Saxon invaders. Wales was at this time populated by Celts.
In contrast, the New English, the wave of invaders who came to Ireland during the Elizabethan era, kept their English identity, religious, social and cultural traditions and unlike the Normans and the Old English, remained distinct and separate from the rest of Ireland.