This story is important because it illustrates the epic similie. The reader should be able to predict what will happen to Odysseus if he journeys home without the help of the gods. He should befall the same tragic fate as Agamemnon, the great grandson of Tantalus.
Tantalus boiled his son Pelops and fed his mortal flesh to the gods. Enraged, the gods sent him to Hades and punished him by putting him in a pool whose water receded each time Tantalus tried to get a drink near a tree with boughs that swayed away each time Tantalus reached for a fruit.
Tantalus' son Pelops had two children, Atreus and Thyestes.
Atreus' sons are Agamemnon and Menelaus.
Menelaus marries Helen who is stolen away by Paris to Troy which is why the Acheans sail to Troy for the Trojan War (basic plot of the Iliad).
Agamemnon's wife is Clytemnestra. Their children are Iphigenia, Orestes, and Electra.
Thyestes stole Atreus' wife. In return Atreus killed two of Thyestes' sons. Thyestes' surviving son, Aegisthus, courts Clytemnestra while Agamemnon is off fighting the Trojan War for almost ten years.
Upon returning from Troy, Agamemnon is killed by Aegisthus & Clytemnestra. He is avenged by his son Orestes.
Odysseus meets Agamemnon in the Underworld in Book 11. He learns that Agamemnon's wife betrayed him and Agamemnon warns Odysseus of returning home to Ithaca to meet a similar fate.
There are two versions of Tantalus' myth. Either Tantalus stole ambrosia, the food of the goods, and thereby revealed the gods' secrets to men; or Tantalus tricked the gods into eating human flesh. Tantalus' punishment was to stand in a pool of water, near to a fruit tree, constantly hungry and thirsty. Whenever he reached for fruit, the branches moved away from his grasp - whenever he tried to drink the water in which he stood, it drained away. His punishment is memorialised in the verb "to tantalise".
Sisyphus violated the guest-host relationship so important to the Greeks by killing guests in his house - he was also famed as the craftiest and most sly of men. His punishment was to roll a huge boulder up a hill, only for it to roll all the way back down after it reached the top.