What is the significance of the following passage:"We will establish our estate upon Our eldest, Malcolm, whom we name hereafter The Prince of...

In these lines, King Duncan is naming his oldest son, Malcolm, to be the next king of Scotland. The only problem is that Macbeth has been promised the kingship by the witches in the first scene of the play. Part of the witches' prediction has already come true. Duncan has just named Macbeth as Thane of Cawdor. In his haste to be king, Macbeth thinks that Duncan may also name him his heir. Instead, Duncan chooses Malcolm for that honor. This choice provides the entire motivation for the rest of the play. As Macbeth says,

"The Prince of Cumberland! That is a step

On which I must fall down, or else o'erleap,

For in my way it lies."  (I, iv, 55-57)

In other words, since Malcolm has been named heir to the throne, Macbeth must figure out a way to get Malcolm out of the way so that Macbeth can be king, as the witches have foretold.