In Hamlet, Claudius has killed the king (his brother and Hamlet's father). It isn't long before Hamlet knows that "something is rotten" and realizes that has happened. Claudius knows that Hamlet is aware of what he has done. As a result, Claudius knows he has to get rid of Hamlet.
At this point in the play, Hamlet has killed Polonius, driving his daughter, Ophelia, insane. Laertes, the son of Polonius, wants revenge. Hamlet has fled. Laertes arrives on the scene and demands to know what is going on. He is in a rage and likely to kill anyone he suspects of having a hand in destroying his family.
Claudius knows that he can turn Laertes' rage to his advantage. He quickly says that getting revenge is a good thing -- so long as he gets revenge on the right person. Claudius begins subtly manipulating Laertes, who does not immediately know who killed his father. He nearly thinks Claudius is guilty, but Claudius begins to manipulate him immediately.
Claudius is hoping that Laertes will take care of Hamlet for him. In this way, Claudius can be rid of Hamlet and Laertes can have his revenge. Claudius begins by calming Laertes down. He explains that Hamlet is the guilty party, and then explains why he covered for Hamlet and did not punish him for the crime of killing Polonius.
Claudius explains that he could not have executed Hamlet for the crime because it would have upset the queen (and the people) too much. Laertes, he says, has a right to avenge his father's death. Claudius explains that a duel is the perfect way for Laertes to get his revenge. Laertes agrees to use a sharp sword and even recommends applying poison to the end so even a scratch will kill. If those things don't work, Claudius will give him a poisoned drink after the duel.
In this way, Claudius has manipulated Laertes into dueling Hamlet. In this way, Laertes can avenge his father's death and Claudius can get rid of Hamlet without appearing to have a hand in it at all.
Conceited means that someone is very proud or vain. The first one says that he communicates intimately with himself and when he agreed on something it was done.
