What are Marcellus and Bernardo trying to convince Horatio they see on their watch?
In Shakespeare's Hamlet, the two guards are trying to convince Horatio that they've seen a ghost. Horatio doesn't believe them, until the Ghost actually appears.
Horatio believes their ghost story is just a "fantasy." Bernardo, still trying to convince him, says:
Last night of all,
When yond same star that's westward from the pole
Had made his course t' illume that part of heaven
Where now it burns, Marcellus and myself,
The bell then beating one-- (Act 1.1.33-37)
He is about to tell the story of the Ghost's appearance the previous night, when
And Horatio needs no more convincing. The Ghost appears at the same time during the night as it did the previous night.
What you are talking about here can be found at the very beginning of the play. The two guards are trying to convince Horatio that they actually have seen a ghost. Of course, they are telling the truth. They have been seeing a ghost -- the ghost of King Hamlet. Horatio will see the ghost as well and will then get Prince Hamlet to come and see the ghost as well.
Once Prince Hamlet sees the ghost, the main action of the play will start because the ghost will tell him that his uncle Claudius killed his father.