Gaelic is a Celtic language, like those spoken in parts of Scotland and Wales. Celtic speaking peoples populated most of the British Isles before, during, and after Roman occupation, having originally migrated from northwest Europe where Celtic languages and Celtic culture also dominated prior to Roman conquest.

In Britain, the arrival of the Anglo-Saxons in the 5th and 6th centuries pushed many of the Romano-Celtic peoples into western kingdoms like Wales, Cornwall, and Strathclyde.

Ireland had never been conquered by Rome. The language and culture there remained Celtic, later heavily influenced by Scandinavian language and culture in the 8th and 9th centuries.

Modern Irish people largely speak English because of conquest and colonization by England throughout the Medieval period, but Irish Gaelic is also spoken (last time I was in Ireland it was used in road signs) and Gaelic culture remains a part of the Irish identity.

edit: spelling

For an insight into the people, I'd suggest O'Malley's Origins of the Irish. His hypothesis is that the original (hunter-gatherer) Irish came from an area, now partially submerged, around the Isle of Man and south-west Scotland. The farming population came from Britain and north-west France. The 'Gaelic' Irish were probably a Celtic elite fleeing Britain and north-west France from the Romans.

As for the Gaelic language, it's a 'Celtic' language like Breton, Welsh and Gaulish, but it probably arrived in the late Bronze Age.

Do you mean the Irish language or the Irish people?

Finally, something I know! The early Irish myth the Lebor Gabála Érenn ("book of the conquests of Ireland" usually called the Book of Invasions) gives a mythical account of the Gaels as a lost tribe from Israel, fitting the Christian monk's agenda (the people who wrote Irish myths). After multiple races invade and are cast out the human race arrives known as the Milesians who battle the local pagan gods and cast them into the Otherworld. People usually agree that the Milesians are the Gaels, but the book says they came from Spain. It's almost certain that the Gaels, as a Celtic-speaking people, came from Gaul (modern France and the surrounding areas). The Celts arrived in the British Isles from Gaul around 500 BCE and individuated into the Brythonic and Goidelic speaking Celts sometime then, where the former would settle England, Wales and the east of Scotland and the latter, the Gaels, would settle Ireland, the Isle of Man, and the east of Scotland, shortly after or within the next century of the settlement of Britain.