What is the climax of "The Little Mermaid"?

The climax of "The Little Mermaid" occurs when she must choose between killing the prince to save her own life or sacrificing herself so he may live.
The climax of any story is the moment of greatest narrative tension. Usually, this involves the protagonist squaring off against the antagonist. This might be an individual that opposes their goals (usually called the villain) or an event they have to face. However, sometimes the antagonist is internal, something within the protagonist that they must overcome.
In "The Little Mermaid," the mermaid's antagonist turns out to be her own desire for the prince and an immortal soul. She wants this more than anything, to the point where she's willing to give up her voice, leave her family behind, and suffer great pain to achieve these goals. When the prince decides to marry another woman, these goals will be lost to her forever.
During the climax, the mermaid gets the chance to return to the ocean with her sisters on the condition that she stab the prince in the heart with an enchanted dagger. If she refuses, she will turn to sea foam and die without an immortal soul. The mermaid considers killing the prince, which creates suspense for the reader. However, her love for him overcomes her selfishness, and she chooses to die for him instead.
After that, the narrative tension winds down. The mermaid gets the chance to earn a soul with the help of the daughters of the air, which makes up the falling action. The resolution is that she gains a soul that will live on forever after.

In Hans Christian Andersen's "The Little Mermaid," the title character is motivated, out of love for the prince and out of a desire to attain an immortal soul (which mermaids do not possess), to make a deal with the sea witch to attain human legs so as to coexist with humans. As payment, the sea witch takes her tongue, removing her ability to speak or sing. Should the prince marry her, she would stand to receive an immortal soul, but should the prince marry someone else, then she would turn into sea foam.
Ultimately, in Andersen's tale, the mermaid's romantic hopes end in disappointment. While the prince has great affection for the mermaid, he has no desire to marry her, and by the end of the story, he is set to marry another. This brings about the climax of the tale, by which, after all her agony and torment, the mermaid finds herself face-to-face with her own mortality, aware that she will turn to sea foam with the coming morning.
It is in this context that the sisters return to her, having made their own deal with the sea witch, bartering their hair for a chance to save the mermaid's life. They present her with a knife, and tell her that if she were to drive the knife into the prince's heart, killing him, then she would return to her life as a mermaid. This is the dilemma that stands at the center of the story and serves as its key scene: she can kill the prince to save herself, or she can spare the prince, which means an immediate death with no hope of salvation. She tosses the knife away, surrendering to that fate.
The story ends on a hopeful note, however, as because of her suffering and sacrifice, she is transformed once again, not into a mermaid but into one of the "daughters of the air," who are given the chance to earn their souls through good deeds and service to the world.

The climax to The Little Mermaid (the movie) begins when Prince Eric and Ursula head out on the ship to get married, and Ariel decides to stop the wedding. In the scuffle that takes place, Ursula's spell over Ariel and Prince Eric is broken. He gets his mind back, and Ariel gets her voice back. They run toward each other and kiss. Unfortunately, the kiss comes a few seconds too late, and Ariel transforms back into a mermaid. Ursula then drags Ariel with her back into the ocean. Prince Eric heads out in a row boat to try and save her, and King Triton takes action as well. He overwrites Ariel's name on the contract and gives himself up in return. Ursula then becomes extremely huge and creates an enormous storm and whirlpool in the ocean as she taunts Ariel and just about anybody else who dares challenge her. Prince Eric then takes control of a ship that has a very pointy tip and steers it into Ursula's gut. This kills her and ends the storm. From this point forward, the story moves toward its resolution. Triton's power is restored to him, and he transforms Ariel into a human. She runs to Prince Eric, they get married, and they live happily ever after.