In Macbeth, who is named Prince of Cumberland?
Malcolm, King Duncan's son, is proclaimed Prince of Cumberland in Act I Scene 4. Macbeth is greatly perturbed by this news as it means there is another obstacle - other than Duncan - who stands in his way of accessing the throne as the witches have prophesied.
The Prince of Cumberland! That is a step
On which I must fall down or else o’erleap
For in my way it lies. Stars, hide your fires;
Let no light see my black and deep desires.
The situation is appeased for Macbeth in the short term as Malcolm flees to England and his brother Donalbain to Scotland, after the murder of Duncan is discovered. This leaves the way clear for Macbeth to be proclaimed the new king.
In the play Macbeth by William Shakespeare, it is King Duncan's son Malcolm who is to be the next prince of Cumberland. The title is important because of the region it alludes to. Cumberland used to be a large county in North Western England (now it is part of the new area called "Cumbria") which matters because it borders on some of the lands mentioned in the play. Alongside it are Northumberland and Durham and to the North ... is Scotland. So anyone with these lands assigned to him would have a strengthened hand. It is important to remember that although in England regions were appearing to settle down into "governed" areas, in Scotland tribalism and clan rebellions went on for much longer.
In Act I Sc.4 King Duncan announces that his son, Malcolm will be the new Prince of Cumberland. Macbeth sees Malcolm as a threat to what he now takes seriously as his destiny to become King of Scotland. This marks a major turning point in Macbeth's changing morality. Macbeth makes this clear by famously asking in an aside (private speech), for the stars to hide their light least they reveal his dark and evil purpose to kill King Duncan:
the successor of the king is malcolm hi son!
Just to add some historical background as a follow-on from the answer here from "coachingcorner", the title Prince of Cumberland in early medieval times was as significant in Scotland as the title Prince of Wales is in today's United Kingdom. The title of Prince of Wales is given to the first-born male heir to the British throne in the same way as the Prince of Cumberland was the heir to the Scottish throne. In this respect the territory of Cumberland did indeed hold great significance, but as the southern extremity of the then Kingdom of Scotland and not the northern extremity of Anglo-Saxon England.
At the time of William the Conquerer in England, Cumberland (and the wider area now known as Cumbria) was a significant part of Scotland, bordering as it did the Anglo-Saxon kingdom to the south. The last "King of Cumbria", Dumnall (corrupted to Dunmail as in Dunmail Rise near Grasmere in Cumbria, anglicised to Donald) was Domnall mac Donnchada (Donald, son of Duncan) or Donald III, the last monarch of the Royal House of Canmore (from the Gaelic Ceann Mor, meaning "Great Head" or "Great Chief"), the last Gaelic monarch.
In the modern era the title Duke of Cumberland was given to the younger son of the Hannoverian monarch George II, Prince William. At the Battle of Culloden in 1746 he became known as the Butcher of Cumberland for the brutality he showed in his command of the forces which killed off the Jacobite Uprising. As a consequence the use of Cumberland in royal titles eventually fell out of favour and so the ancient link between Cumberland (Cumbria) and Scotland was lost officially. Today Cumbria is a county within England but there are some, myself included, who are proud of our ancient Scottish heritage and feel a greater affinity with our cousins north of the border than we do with the rest of England.