In the play Hamlet, what is the significance of Act 1 scene 1, and what would the play be like without it?

Act 1 Scene 1 of any play, especially a Shakespearean play,
establishes the tone of the play and the background situation of the setting. It is important to
remember that in Shakespeare's theatre, the Globe, there are no lights to dim to signal the start
of the play, so the characters would need to enter center stage and just start talking.
Shakespeare usually opened his plays with minor characters so that the audience was primed and
ready for the revelation of the major characters and the major conflicts of the play by scene 2.
The theatre would have settled down and been ready for the big story by then.

In this play, scene one has four minor characters, guards of
Elsinore castle and Hamlet's best friend Horatio, talk about the strange occurrence they have
seen two times before. The guards report to Horatio that they have seen a ghost that looks like
the late King Hamlet. Horatio, as a university student and a scholar, is brought out to make sure
they aren't just imagining things. At first Horatio is skeptical, but once the ghost actually
appears, he is a believer. He attempts to command the ghost to speak, but the ghost merely
appears and then disappears again. The men decide that they MUST tell Hamlet the disturbing news
that a ghost like his father wants to see him.

The scene also serves
as a means for the Horatio to explain to the guards why there has been a recent increase in
military activity around the castle. Horatio reports that the young Prince Fortinbras of Norway
has "sharked up a list of lawless resolutes" to attack Denmark in an attempt to regain lands that
his father lost to King Hamlet several years before.

By the end of
the scene, we have established the external conflict with Norway and the internal trouble that is
suggested by the appearance of the ghost. A Renaissance audience would have believed that the
ghost comes for some terrible reason -- never anything good. This scene is essential to
establishing this background information. Once the audience has all of this information, they are
ready to meet Claudius, Hamlet, Gertrude, Polonius and all of the other significant characters in
the next scene.