One can not mention a swan in connection to Hellenic mythology and not think of the love affair between Zeus and Leda, the affair that led to the birth of some very influential people in Hellenic mythology. Leda (Λήδα) was the daughter of the Aetolian king Thestius (Θέστιος), and wife of the king Tyndareus (Τυνδάρεως), of Sparta. Zeus looked upon the beautiful Leda and fell for her instantly. In the guise of a swan, He came to her, seeking refuge in her arms from an eagle. Leda sheltered Him, and lay with Him--either after He transformed into a man, or while He was still a swan. That night, she also slept with her husband. She became pregnant and gave birth to two eggs, one housing Helene (Ἑλένη) and Klytaemnestra (Κλυταιμνήστρα), and the other Kastor and Polideukes (Κάστωρ καὶ Πολυδεύκης). Its also said that Zeus laid with Nemesis and She gave birth to an egg that housed either Helene alone, or her sister as well. A shepherd found the egg and took it to Leda, who hatched it, and adopted the child or children. In the first case, the swan is Zeus, in the latter, the swan was placed in the sky to celebrate the birth of Helene.
The second male the constellation is identified with is Orpheus. After being forced to leave Euridice in the Underworld, he travels the world with his lyre. He renounces both women, and many of the Theoi, pained as he is by the loss of his beloved wife. One day, he either sacrifices to Apollon--one of the few, or even the only Theoi he still offers to--at a shrine to Dionysos, and is discovered by the female revelers. Alternatively, the revelers stumble upon him as he plays, and they can't appreciate his divine music, or a group of women falls upon him for denouncing women. Whatever the case, Orpheus is ripped apart. His lyre is placed into the sky, and he is placed in the sky near the lyre by Zeus in the form of a swan. Plato explains this odd choice in his 'Republic', when he speaks of reincarnation:
"He said it was a strange, pitiful, and ridiculous spectacle, as the choice was determined for the most part by the habits of their former lives. He saw the soul that had been Orpheus’, he said, selecting the life of a swan, because from hatred of the tribe of women, owing to his death at their hands, it was unwilling to be conceived and born of a woman." (X, 620a)
I'm going to conflate the last four. All of them were named 'Kyknos' (Κύκνος). The first was a bloodthirsty son of Ares, who slaughtered all his guests when they came to his door. Hēraklēs killed him, either in self-defense, or in honorable battle. In one version of the story, Ares transforms His son into a swan before Hēraklēs can deliver his deathly blow, preferring this solution over his son's death. The second Kyknos was the son of Poseidon, and King of Kolonai. As the son of Poseidon, he was impervious to both spear and sword attacks, as such, he was suffocated by Achilles during the battle for Troy. After his death, this Kyknos, too, was turned into a swan.
The third Kyknos was a human King of Ligûria, and friend--or lover--of Phaëthon, son of Hēlios. When Phaëthon was killed by Zeus after scorching the earth, Kyknos was inconsolable. He spent the rest of his life mourning Phaëthon. To relief his suffering, Zeus transformed him in a swan, and was later put into the sky by Apollon. The last of the Kyknos' associated with this myth was a son of Apollon. This Kyknos was a handsome but arrogant man. Many young boys fell for his looks, but his personality drove them off again. One of the young men, Phylios, loved Kyknos unconditionally, but Kyknos felt the need to test Phylios' resolve, trying to scare him off. The first task was to kill a lion that was threatening the neighborhood without use of any weapons. The second task was to catch two man-eating vultures of enormous size that were posing an equal threat to the neighborhood, again without use of any devices. Finally, Phylios had to bring a bull to the altar of Zeus with his own bare hands. With divine help, Phylios managed to complete all three tasks, but Hēraklēs cured the boy of his love for Kyknos. Kyknos, enraged and humiliated, took his own life by drowning. His mother did the same. Out of love for the both of them, Apollon turned them both into swans. In the words of Roman poet Ovid, in his Metamorphoses:
"Whilst here, within the dismal gloom, alone,
The melancholy monarch made his moan,
His voice was lessen'd, as he try'd to speak,
And issu'd through a long-extended neck;
His hair transforms to down, his fingers meet
In skinny films, and shape his oary feet;
From both his sides the wings and feathers break;
And from his mouth proceeds a blunted beak:
All Cycnus now into a Swan was turn'd,
Who, still remembering how his kinsman burn'd,
To solitary pools and lakes retires,
And loves the waters as oppos'd to fires."
The next time you look to the heavens and see constellation Cygnus, you'll have quite the stories to tell. The constellation is visible at latitudes between +90° and −40°, and best visible at 21:00 (9 p.m.) during the month of September.