Cronos has written exactly what I, as a Scot, was about to.
The only thing I would add is that whilst Celts from Ireland came across about 500AD and settled in the SW and W Highlands, the people we know as the Scotti/Scots, recent study has shown that the widely believed mass migration didn’t happen, and that most Celts in the area were already there, King Fergus and his Scotti/Scots came over in far less numbers than most people think.
http://www.electricscotland.com/history/articles/scotsirish.htm
http://www.gla.ac.uk/schools/humanities/staff/ewancampbell/
http://senchus.wordpress.com/2008/08/20/scottish-origins-myths-and-misconceptions/
http://irishtribesman.blogspot.co.uk/2005/04/scots-did-not-come-from-ireland.html
I did not know that there was no mass migration — from what I’ve read (Norman Davis’ Lost Kingdoms), the place that the Scotti came to was cleared of Picts and had been abandoned by the Romanized Britons (the land between the Antonine and Hadrian walls)