In Greek mythology, Telemachus (ancient Greek Τηλέμαχος Têlémakhos, translated as one who fights from afar, suggesting that he is an archer) is the son of Odysseus and Penelope, and a character in the Odyssey. 
In search of Odysseus
Telemachus was still a child when his father marched into the Trojan War, and in his nearly twenty years' absence he grew to an adult. After the gods in assembly had decided that Odysseus should return home from the island of Ogygia, Athena, assuming the appearance of Minds, king of the Taphians, went to Ithaca and advised Telemachus to drive the troublesome suitors out of his home. of his mother and went to Pylos and Sparta in order to gather information about his father. 
Telemachus followed the advice, but the suitors refused to leave their home, and Athena, still in the form of Mentor, accompanied him to Pylos. There they were received hospitably by Nestor. His son Pisistratus slept next to him, while Mentor did so on the ship with the crew. The next day, a daughter of Nestor, Policasta, bathed Telémaco. Then Pisistratus led him to Sparta. Menelaus also received him kindly, and communicated to him Proteus' prophecy about Odysseus: that he was being held against his will by the nymph Calypso, who loved him. 
Return of Telemachus
From Hyminepatre, Telemachus returned home, and once there he found his father with his faithful swineherd Eumeo. However, Athena had transformed Odysseus into a beggar, so Telemachus did not recognize his father until later, when he revealed his identity to her. Father and son then agreed to punish the suitors; and when they were finished with or dispersed, Telemachus accompanied his father to see his elderly grandfather Laertes. 
In post-Homer traditions, it is recorded that, when Odysseus feigned stupidity in order not to go to the Trojan War, Palamedes put the little Telemachus before the plow with which Odysseus labored to plow the beach and sow salt.     According to some sources, Telemachus was the father of Persepolis and the poet Homer with Policasta (sometimes called Epicasta), daughter of Nestor, and of Ptoliportes with Nausícaa, daughter of Alcinoo. [ 9]   Others say that he was induced by Athena to marry Circe, and that he was the father of Latino with her.  Or that he married Casífone, her daughter, but in a dispute with his mother-in-law killed her, so that in turn Cassipone killed him.  One version states that Odysseus, because a prophecy warned that his son was dangerous to him, banished him from Ithaca. Servius makes Telemachus the founder of the city of Clusium in Etruria. 
Smith, W., ed. (1867). "Telemachus". A Dictionary of Greek and Roman biography and mythology. Boston: Little, Brown & Co. iii. 989. OCLC 68763679.
Notes and references
Wikimedia Commons hosts a multimedia category on Telémaco Odyssey: Spanish text on Wikisource:
Telemaquia: Cantos I - IV.
Telemachus back in Ithaca: XV - XXIV.
Odyssey: Greek text on Wikisource.LAMB, Charles: The Adventures of Ulysses, Chapman & Hall, 1808; in English.
Reproduction, in electronic facsimile, on the Internet Archive site.
VIII: The transformation from king to beggar - Eumeo and the shepherd - Telemachus.
IX: The queen's suitors - The beggars' fight - The gathering of weapons - The meeting with Penelope.
X: Insanity Infused From Heaven - The Arch of Ulysses - The Carnage - The Conclusion.BULFINCH, Thomas: Bulfinch's Mythology.
I: The era of the fable or Histories of gods and heroes (The Age of Fable or Stories of Gods and Heros, 1855).
I, 30: The Phaeacians; The Fate of the Suitors (The Phæacians - Fate of the Suitors): text