Possibly the most famous mermaid story of them all because of the recent Disney movie. Interesting trivia: the name Ariel was made up for the Disney film, the protagonist in this mermaid story is never given a name.
The little mermaid lived under the sea with her five sisters, her father the Sea King, and her grandmother. Of the six girls the little mermaid was the youngest and prettiest. From a young age she had been interested in the world above the waves but she was too young to go there. The mer-folk, I should say, live to be very old, perhaps 300 years or more, but they have no souls. Once dead they can never be reborn but change into sea foam. To gain a soul they must win the love of a human. However, most mer-folk do not desire souls, they feel they are superior and better off than the poor humans.
On the little mermaid's fifteenth birthday she was allowed to visit the surface. There she saw a ship and aboard it a handsome prince. A terrible storm was raging and the ship was destroyed, but the little mermaid rescued the prince and brought him safely to the shore. He was unconscious, however, and never knew who or what had rescued him.
The little mermaid had fallen in love with the prince and wanted to have him for her husband. She visited a Sea Witch who laughed at her desire to be human but offered her a trade. In exchange for the mermaid's voice she would give her legs. The mermaid hesitated, but love for the prince and the desire to have a soul won out. She accepted the Sea Witch's bargain.
The little mermaid swam to the palace of the prince and climbed up on the stairs. There she drank the Sea Witch's potion and swooned. Later she awoke to find the prince standing over her. She had two beautiful legs now but she was entirely naked. She wrapped her long thick hair around herself. The prince asked her name and where she was from but of course she could not tell him.
The prince was enchanted with the lovely mute and they spent all their time together. The prince, however, never fell in love with her. He pined for a maiden he had seen at the shore after his ship had been wrecked. More, he was supposed to marry a princess from a neighboring kingdom.
The prince was obliged to see the princess, though he had made up his mind he would never marry her. All that changed when he actually met her and saw that she was the maiden he had seen on the shore. The little mermaid was broken-hearted but could not speak to tell the prince who she was and how much she loved him.
When the prince and princess married the little mermaid jumped into the sea hoping to die. Instead she was transformed into a daughter of the air. In this form she could finally obtain a soul, although to win it she had to perform good deeds for hundreds of years.
More mermaid story trivia: in Shakespeare's play "The Tempest" Ariel is the name of a creature called an "airy spirit".