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The story 
Elpenor was not especially notable for his intelligence or strength, but he survived the Trojan War, and appears in the Odyssey. He is the youngest man to survive the Laestrygonians. While Odysseus was staying on Aeaea, Circe's island, Elpenor became drunk and climbed onto the roof of Circe's palace to sleep. The next morning, waking upon hearing his comrades making preparations to travel to Hades, he forgot he was on the roof and fell, breaking his neck. Odysseus and his men apparently noticed his absence, but they were too busy to look for him. When Odysseus arrived in Hades, Elpenor was the first shade to meet Odysseus, and pleaded with him to return to Aeaea and give him a proper cremation and burial. After finishing his task in the underworld, Odysseus returned to Aeaea and cremated Elpenor's body, then buried him with his armour and marked the grave with an oar of his ship.
Elpenor in Homer's Odyssey 
When Odysseus is eventually convinced by his shipmates to leave Circe's island, Aeaea, a still-drunk Elpenor is woken up in the morning by the sound of his crewmates packing up and leaving. Trying to get down from the roof of Circe's stone house to join his comrades, the drunken Elpenor missed a step on the ladder and fell to the ground, breaking his neck. Because his crewmates are in such a hurry leaving the island - they had been there for about a year already - they do not notice Elpenor's absence. Later in Book XI, when Odysseus is making a sacrifice to meet the shade of Tiresias, he is surprised and upset to see Elpenor's shade, so he calls out to him and asks him how he came to be there. Elpenor tells the same story as described above, before begging Odysseus to give him a true burial. He pleads for Odysseus to return to Aeaea to find his body so that he can be cremated, and then buried, in his full set of armor. Finally, he asks to be given an anonymous burial with the oar he pulled to mark his grave. This is because he wants to be remembered as a sailor, as opposed to a drunkard who died a strange and dishonorable death. This is a clear example of the Greek belief in Kleos, or honor, in their daily life: for the Ancient Greeks, it was always better to die young with honor as opposed to surviving to old age but without accomplishing much. Clearly, Elpenor is embarrassed to have died young without any honor, and seeks to hide his true fate from everyone else.
Later historic uses 
The story of Elpenor can be described as a mirror to the story of Palinurus in Virgil's The Aeneid. In the Aeneid, Palinurus, one of Aeneas' men, falls overboard and ends up swimming to an island nearby. He is killed on the island by the natives that live there. Later on in the story, Aeneas travels to the underworld where he sees Palinurus. There, Palinurus pleads with Aeneas to give him a proper burial.
Modern uses 
The character of Patrick "Paddy" Dignam, whose funeral is the focus of Episode 6 ("Hades") of Ulysses by James Joyce, is a modern counterpart to Elpenor. This chapter of Ulysses is a main inspiration for the film Bye Bye Braverman.
Elpenor is the subject of the short novel Elpénor by Jean Giraudoux, published in 1919, which retells some of the stories of the Odyssey in humorous fashion.
Also, Derek Mahon suggests Elpenor (but does not name him specifically) in his poem "Lives." Mahon talks of a decaying oar, planted in a beach, thinking of Ithaca.
Pound also makes use of Elpenor in the first of his Cantos: 'But first Elpenor came, our friend Elpenor / Unburied, cast on the wide earth, / Limbs that we left in the house of Circe, / Unwept, unwrapped in sepulchre, since toils urged other.'
- Odyssey, Book X
- Fitzgerald, Robert, trans. "Book XI." The Odyssey. New York: Farrar, Straus and Giroux, 1998.