What is "rotten in the state of Denmark," as Marcellus tells us? What do we learn about the situation in Scene 1? In Scene 2?
What is rotten in the state of Denmark is regicide, the killing of a king. It was believed in Elizabethan times that this was one of the worst possible types of murder - it upset the very fabric of existence. Plus, in this case, not only has the king been murdered, but his brother is the murderer and that very brother has now married the dead king's widow!
In Scene I, Horatio, Marcellus and Bernardo are standing watch and they see the ghost of Hamlet's father, the dead king that has been killed by his brother, Claudius, who is now king. In Scene II, Hamlet is talking to his mother, Gertrude, and his uncle, Claudius, who is now king. Hamlet's father is dead and within a very short period of time, Claudius has married Hamlet's mother. Hamlet is very depressed by this, thinking that his mother has not honored his father's memory by marrying so soon. Gertrude and Claudius are worried about Hamlet's mental health. Later, Horatio, Marcellus and Bernardo come in and tell Hamlet that they have seen the ghost, so Hamlet resolves to watch for the ghost himself that night.
You can read these scenes right here on eNotes.
There is something going on after the King dies, it isn't clear what exactly it is, but the fact that a ghost has visited the castle now on more than one occasion suggests to Marcellus and to others that something has gone terribly wrong.
In scene i, we learn about the ghost and that it resembles Hamlet Sr., the former king. By his appearance it appears that he is not exactly pleased about the whole situation but they can't figure it out.
In scene ii, we learn that Claudius has just married Gertrude and assumed the throne. We learn that Hamlet is quite perturbed about things, but we don't know exactly what yet. We also learn a little bit about some problems with young Fortinbras and Norway that will come into play later.