This spooky ghost scene sets the mood for the rest of the play, with Fransisco describing a supernatural cold that chills the very bones. The appearance of the ghost introduces a key theme of the play - how do we distinguish between appearance and reality? Is the ghost really there? It is this question that will haunt Hamlet for the rest of the play. Barnardo and Fransisco's conversation also introduces the political unrest that is in Denmark through the change of king, which is aptly mirrored by the change of guard that is progressing. The "sickness at heart" that Fransisco feels evokes the wrongness of what is progressing in Denmark, with the rightful heir returning, but being deposed.
We are also introduced to the character of Fortinbras, who is a foil for Hamlet. We find out that he is the son of a conquered father, who is duty-bound to revenge that defeat by re-taking the conquered territories. This also introduces another theme key to the play: that of the relationship between fathers and sons and the sons' responsiblity to obey and avenge their father whatever the cost.
Horatio's fear of the ghost and his questioning of him is devised to try to work out what this apparition is. There are a number of possibilities that are considered by characters throughout the play, but the fact that the ghost is wearing armour causes Horatio to treat it as a portent or a prediction of what is to come.
This scene then introduces Elsinore as a kind of prison, victim to supernatural and political forces out of its control and beyond its range. A rather unstable place to be!