The old Goidelic name for Britain, retained today as the Ghallig name for Scotland
First of all, 'Alp' is believed to be an old Celtic word for a rock or crag (Cf the modern Irish 'ailp' which means a large mass or lump of anything) and hence therefore the Alps to refer to that mountain range in Switzerland.
From 'Alp' was derived Alpa or Alba, the 'hilly country', the Goidelic name for Britain, perhaps based on the appearance of the traditional cliffs of Dover when viewed from the coast of Gaul. Since the Goidelic branch of the Celtic language was spoken not only in Ireland also in Argyll in the south-west of modern Scotland, the Goidelic 'Alba' for Britain was how these eastern Gaels referred to where they lived. Gradually, over time, the meanining of 'Alba' shifted so that it began to mean not Britain as such, but that specific part of Britain inhabited by the Goidelic speaking Scots of Argyll, or the kingdom of Dal Raida as it became known as between the sixth and ninth centuries.
By the mid to late ninth century, with the ascent of a new line of kings descended from the ruler Kenneth mac Alpin who succeeded in uniting Dal Riada with the neighbouring kingdom of the Picts, the name of Dal Riada disappears from the comtemporay annals and is replaced by references to the 'Kings of Alba'. The transformation is complete, and 'Alba' becomes 'Scotland'.
Alba was often rendered into Latin as Albania, which is obviously a potential source of great confusion these days with the modern state of that name located in the Balklans, and therefore an entirely archaic usage, but which was further rendered into English as Albany, which remains an old and poetic synonymn for Scotland.
The modern Welsh word for Scotland is 'Alban', although Scotland is always referred to as 'yr Alban' or 'the Scotland', for some strange reason. Whether the Welsh 'Alban' comes from the Goidelic Alba or is derived from the same common Celtic root of 'Alp' is not known. Since the Welsh for Britain is 'Prydein' and is derived from the Latin Britannia, which itself is derived from the Greek word for the island, it is obviously possible that the Brythonic inhabitants of Romanised Britain began using a derivative of the Latin name for their homeland in preference for some common Insular Celtic word such as 'Alpa'. The Brythonic language may therefore have followed a similar path to the Goidelic, with the meaning of an archaic Alban transferring itself from 'Britain' to 'Scotland'.
Which may explain the derivation of the word Albion which is another poetic name (as in Perfidious Albion), this time for Britain, or sometimes depending on context, to England alone. Although Albion is often alternatively conjectured to come from the Latin albus for 'white', a specific reference to those white cliffs of Dover, it is much more likely that it comes from the same common 'Alba/Alban' root.
It is also worth mentioning that other Alban or Albanus in the Latin, the first Christian martyr of Britain and after whom the town of St Albans is named. No one quite knows where the Latin name of Albanus comes from, but if the suggestion that there was once some common Insular Celtic word of 'Alba/Alban' for Britain is correct, then what better name could there be for the very first British Christian martyr than a name that meant Britain itself?.
For wertperch who asked the question.
SOURCES
Christopher Snyder An Age of Tyrants (Penn State Press, 1998)
WA Cummins The Age of the Picts (Sutton, 1995)
Together with various dictionaries;
Oxford English Dictionary
The American Heritage Dictionary of the English Language, Fourth Edition
Webster's Revised Unabridged Dictionary