What is the exposition in Act I, Scene 1?
An expository passage in literature is one that "sets the scene," in the sense that it provides the necessary context at the beginning of a work for the reader to be able to follow what then happens. In this play, as in many Shakespeare plays, there is an expository conversation in the opening scene, in this case between Horatio, Marcellus, and Barnardo.
Through their discussion of, and the appearance of, the Ghost, these characters explain to the audience both what has been happening in Denmark more broadly—King Hamlet has just died; Fortinbras has declared war on the Danes—and in the locality: people have been muttering about seeing a Ghost. Barnardo's thoughts as to whether the Ghost could be a portent of doom also invite the audience to wonder whether something is going to go badly wrong for Denmark.
In literature, the exposition is defined as a passage where we are given the background to the story that is about to unfold. Given that definition, you can probably tell that the exposition in this scene comes in the conversation between Horatio and Marcellus.
In this conversation, they go over what is going on in Denmark during this time. They discuss how the late King Hamlet had defeated the Norwegians and how now the new king of Norway was going to attack Denmark. This is the background to the story that is still to come.
In a sense, you can also say this is exposition in that it sort of tells us (though not explicitly) that something is wrong in Denmark. This sets us up for the story of Gertrude and Claudius and Hamlet's dislike of their marriage.