What does the Ghost of King Hamlet symbolize?

The ghost is a symbol of the troubles coming to Denmark. When Horatio, who first disbelieves the guard's tale of seeing a ghost, actually witnesses it with his own eyes, he is unnerved. He states it is a warning that dire events are coming and mentions the way the ghosts rose from their graves in Rome right before the assassination of Julius Caesar. That the ghost appears in full armor is also a fearful sign that bloodshed is in the air. Horatio says:

Such was the very armor he had on When he the ambitious Norway combated

When Hamlet hears the report that his father's ghost appeared in armor, he too fears that "all is not well."
The ghost's appearance is an ominous sign of a world out of joint, foretelling disaster for Denmark. Spirits only walk the earth when heaven and earth are out of alignment.
That the ghost wears armor, and specifically the armor in which he killed the late Fortinbras, also shows the price of violence. Fortinbras's son, also named Fortinbras, is now marching with an army towards Denmark, seeking revenge for his father's death. The cycle of revenge will haunt this play from beginning to end.

The ghost of Hamlet's father is a symbol of the treachery and corruption in the Danish court. Claudius's murder of his brother, the old King Hamlet, initiates a corrupt chain of events that begins with Claudius's incestuous and rapid marriage to his sister-in-law, Gertrude, and progresses through a large number of tragic deaths. There is the death of Ophelia, as well as the murders of Polonius, Rosencrantz, and Guildenstern. Laertes dies, Gertrude dies, and even Hamlet himself dies in the end. Eventually all of the members of the royal family and even their closest advisers must be eradicated; the court of Denmark must be utterly purged in order for the country to move on and prosper again. Though the dead king, himself, was just and good, his ghost can only exist because of the betrayal of his brother. The betrayal and rot spreads until all have been punished and are dead.

The Ghost is a symbol of the consequences of death in the afterlife.
Essential quotation:
“I am thy father's spirit, / Doom'd for a certain term to walk the night, / And for the day confined to fast in fires, / Till the foul crimes done in my days of nature / Are burnt and purged away.”