In Book III, who killed Agamemnon and how was that murder avenged?
In Book III, Telemachus and Athena (disguised as Mentor) have reached Pylus. While there, they learn more about what has happened to Odysseus. In learning this, they also hear about what has happened to Agamemnon.
Agamemnon has been killed by a guy named Aegisthus. Aegisthus did not go to the Trojan War but instead stayed behind and had an affair with Agamemnon's wife. When Agamemnon got back, Aegisthus killed him. Aegisthus then ruled for seven years in place of Agamemnon.
The murder of Agamemnon was avenged by his son, Orestes. Orestes killed Aegisthus in revenge for his killing Agamemnon.
In Book III of the Odyssey, Odysseus' son Telemachus arrives at Pylos to find out more information about his father Odysseus, who has been missing for twenty years, since the Trojan War. From Nestor, the ruler of Pylos, they do learn some things about Odysseus, but are directed to go to Sparta for more information.
Nestor also tells them about the ill-fated homecoming of Agamemnon, the commander of the Greeks during the Trojan War. Agamemnon is the king of Mycenae. While Agamemnon was gone, his wife Clytemnestra was plotting to get rid of him. Clytemnestra is a cunning and ambitious woman who hates Agamemnon. (Backstory: Before the war, the Greek expedition couldn't leave an island because of bad winds caused by the goddess Artemis. In order to appease her, Agamemnon ordered that they sacrifice a virgin girl. This virgin girl just happened to be his own daughter, Iphigenia. They lied to her that she was about to marry Achilles, and then slit her throat. Clytemnestra, of course, will never forgive Agamemnon for killing their daughter.)
Clytemnestra has an affair with Agamemnon's cousin Aegisthus. When Agamemnon returns from the war, Clytemnestra insists that he go take a luxurious bath. Then, Aegisthus kills Agamemnon while Clytemnestra kills Cassandra, a Trojan princess that Agamemnon had stolen back with him.
This murder was avenged by Agamemnon and Clytemnestra's son Orestes, who is furious with his mother for killing his father. Orestes eventually kills Clytemnestra and Aegisthus. Fun family, huh?