The name "Psyche" means "Soul" and her union with Eros (aka Cupid) tells the story of how Love and Soul came to be together. By the way, this story is Roman, not Greek, but it works just as well with the Greek, so that is how I shall tell it. This myth had an enormous impact on fairy tales for the next couple of thousand years.
Long, long ago a king had three daughters. Psyche, who was the youngest of the three daughters, was so incredibly beautiful that people in her village and outlying areas STOPPED praying to Aphrodite, taking Psyche for the Goddess of Beauty instead. That wasn't too good, because Aphrodite got mighty pissed off, and when that happens, you don't want to be on her bad side. She went straight to the source: the innocent Psyche. She grabbed Eros (in this version he is her son) and instructed him to make poor Psyche fall in love with the ugliest man on Earth. Eros, who had done jobs like this on his mother's behalf before, went down to Earth to find her. But when he did he, too, was stunned by her beauty. He was so stunned that as he lay his golden arrow on her heart, he pricked himself and fell in love with her then and there. He was so in love that he erased all of what he had done to her, and went away.
After a while Aphrodite realized that her darling son hadn't quite done his job, for Psyche wasn't falling for anyone, let alone someone hideous. SO Aphrodite sent down a spell of her own on Psyche. As soon as this happened, not another suitor knocked upon their door. Her parents got worried, they wanted their youngest daughter to be a rich noblewoman at least. Psyche's mother, the queen, went off to the Oracle to hear what was wrong.
"Psyche will never marry a mortal. She shall be given to one who waits for her on yonder mountain; he overcomes Gods and men."
Poor parents, poor Psyche! It seemed that she was destined to marry a monster. No one wanted to let her go. Whether because of her physical beauty or the beauty of her soul, everyone LOVED Psyche. Psyche, however, saw the futility of her situation, and knew that she had angered Aphrodite, however mistakenly. She accepted what the Fates had decreed and told her parents to take her to the mountain and leave her for the beast. After many denials, they finally agreed and most of the country accompanied her to her supposed death. As she watched them leave, her heart cried out in sorrow and though she stayed firm with courage she could not stop the tear from trickling down her cheek. Zephyr, the kindly West Wind, saw her sorrow and bore her away to the valley below the mountain into soft grass in place of the harsh rock of the mountaintop. It was there that she woke.
When she woke, she didn't see any monster around her but she did see a lush looking forest. Eager for the protection it could offer she went in. As she entered she heard the sound of water and followed it. Deeper and deeper into the forest she went until she found a bright pool with a fountain, and beyond it a beautiful palace. Then she heard a voice around her, its speaker invisible, telling Psyche that the palace was hers, and the invisible servants around her were to do her bidding. She was delighted and ran all around the palace finding all sorts of wonderful things waiting for her. At last she was tired and hungry and before she could ask a beautiful breakfast was laid out before her. That evening the palace grew dark and a new voice spoke to her. It was her husband! This voice was kind and loving, and she couldn't imagine that it was that of a monster. But that didn't stop her from wanting to see him. She begged him again and again to come in the daylight, but again and again he sadly refused telling her that the day she saw him was the day their happiness came to an end. It was Eros, of course, but he couldn't tell HER that.
Eros was very kind to Psyche in every way that he could be - giving her invisible servants to wait on her every wish - but he never ever let her see his face, ominously warning her that the day she did it would be over. Psyche, though kind and happy with her mysterious husband, was a woman, and with that came an almost insatiable curiosity (according to the Greeks, and the Christians, and most others). She was afraid that she was married to a monster, and wanted to know his true visage. One night she told him that she missed her family, and could they be allowed to visit her, please? Eros heard the loneliness in her voice and agreed, but he knew that this would be their downfall. When her sisters arrived they were very eager to hear about her new life, and asked all about her husband. But when they heard of her arrangement they laughed at her and told her that they had heard she was married to a dragon who was fattening her up now, but would soon eat her. They urged her to take out the lamp one nights and look at him while he slept, carrying a dagger that she might kill him if he was indeed a dragon. At first she held out, remembering the warnings of her husband, but in the end curiosity won out, for she could keep the mocking voices of her sisters from her head.
Finally, one night Eros went to sleep as usual, but Psyche remained awake. She took the oil lamp and lit it looking onto his face. Immediately she recognized his godliness and realized what had been going on. She was filled once again with love and contrition and worship, finally knowing who her husband was. But in her shock her hands trembled and she spilled some of the oil onto her lover. Eros awoke and saw the lamp and Psyche's sorrow and realized what she had done. He gave a cry of grief and then flew out the window. She realized now that she was truly abandoned for as she looked around her the palace had disappeared and she was again in the middle of the wood.
Now begins a different part of the story. Psyche realized what she had done, but she was not about to give up her Love (literally) when she had just truly found it. On her own feet, she traveled to the houses of her sisters, married to their princes, and told the story of her treachery and its penalty. From there she left again, traveling she knew not where, only in search of her husband. At the end of the day she came upon a deserted hall filled with ears of corn and barley and wheat strewn all across the floor. Immediately she began picking up the mess and putting it together in a beautiful and decorative manner, making the deserted hall more like a temple. That is in fact what it was, and as she worked Demeter watched her, smiling at the goodness of her Soul. When Psyche had finished, Demeter appeared before her and said:
Psyche, you are worthy of happiness, and you may yet have it. Go now to the temple of Aphrodite and pray for her forgiveness, perhaps she will reward your patience.
Psyche was astonished that such an important goddess would show her favor, and left at once to do her bidding. She went to the temple of Aphrodite and humbly offered her prayer. But the jealous Aphrodite would not look at Psyche and said that if she truly desired repentence for her sins there was work enough. Saying this she led her into a room full of mixed grain, beans and lentils (the food of doves, Aphrodite's sacred bird), and bade Psyche sort them all into piles before the night was over. There was too much for Psyche to do on her own, but she settled down to do it anyway. As she was working a long train of ants came out of the crack in the wall, and helped her separate the piles. With their help the piles were separated by morning. Aphrodite returned to find the work done, and was even more angry, realizing that her son Eros had helped the girl. But instead of just sending the girl away, Aphrodite gave her some black bread and bade her sleep, saying she would need her wit for her next task. The next morning Psyche awoke to Aphrodite's impatient waiting.
Go now to yonder grove where the sheep with the golden fleece are wont to browse. Bring me a golden lock from every one of them, or you must go your ways and never come back again.
Then Aphrodite left her and Psyche prepared to cross the stream to the grove. But as she waded into the water the reeds swayed and the Naiads called out to her:
"Nay, nay, have a care, Psyche. This flock has not the gentle ways of sheep. While the sun burns aloft, they are themselves as fierce as flame; but when the shadows are long, they go to rest and sleep, under the trees; and you may cross the river without fear and pick the golden fleece off the briers in the pasture."
Psyche thanked the nymphs and did as they bade, and when Aphrodite returned Psyche gave her the fleece she had requested. Aphrodite was more enraged than ever, and cursed her son again for his help. This time she turned on Psyche, thrust a small box at her, and told her to descend to Persephone, the cold Queen of the Underworld, and bring back some of her beauty in the box - for Aphrodite was growing tired in tending her son.
Poor Psyche, she knew what Aphrodite knew, that no human could venture to the Underworld and return. And she realized the Eros must have forsaken her, and held no more value to her own life, turning and preparing to make her descent. But as she prepared another voice whispered in her ear - it was Eros, but she did not know - and told her all the ways to avoid the dangers of the Underworld, and warned her also not to open the box once Persephone's beauty was inside. Psyche did as she was told, and before she knew it she was back in the sunlight on Gaia carrying the box. But as she traveled she thought to herself, Aphrodite does not need the beauty, but how will I please Eros as travel-worn as I am. And so she opened the box.
But the spells of Gods are not meant for mortals and as she opened the box Psyche fell unconscious upon the ground. But Eros had recovered by now and was scouring the countryside for her. Soon he found her, woke her up and bade her return to Aphrodite and wait for him. Happily she did so, while Eros went to Olympus. On Olympus he told the feasting Gods his story, and begged them to appease to angry mother. The Gods, taken with the pure beauty of the tale agreed and summoned Aphrodite, soothing her until she was no longer angry. Then Hermes, the Messenger God, descended to Earth and brought Psyche back with him to Olympus. Once there, the gods bid the shy maiden to take the cup of ambrosia that Hebe held out to her. She took it, and upon drinking it blossomed into the fairest thing you've ever seen.
Eros took her in his arms, and from that day on the two were never parted again.
The original version is in The Golden Ass by Apuleius (4.28 - 6.24), and that's way better than anything I could tell you.