One of the major themes in Shakespeare's Hamlet is that of corruption, rot, and decary. In this play, Shakespeare explores how a spot of corruption can destroy the whole--whether the whole is a person, a family, a court, or a country. Each of these quotes concern this theme. The first quote is in reference to the presence of the ghost. Horatio thinks that the ghost might be appearing because something is wrong in Denmark. The most obvious assumption here is that the ghost is warning of Fortinbras' attack, but we later find out that the "strange eruption" is Claudius' murder of his brother.
The second quote is spoken by Laertes to his sister. He is warning her to remain chaste in her relationship with Hamlet because Hamlet may seduce her but not marry her. He does not want Hamlet to corrupt her virginity, and spoil the flower (Ophelia) before she blooms, or fully matures. The third and fourth ones are spoken by Hamlet himself, and he is commenting on the character of Claudius. Claudius enjoys drinking and partying, and Hamlet contends that a "vicious mole," or fault, such as excessive drinking can corrupt the whole character of a person.
Marcellus, a guard, speaks the last and most famous quotation. He is referring to the appearance of the ghost and its effect on Hamlet. He does not know how true his statement is, but the reader knows that the "something rotten" is Claudius, the king of Denmark and his brother's murderer.