In Greek mythology, Amphitrite goddess of the Sea was one of the fifty Nereids, that is a daughter of Nereus and Doris - that's what Hesiod writes in his Theogony (but Apollodorus says her parents were Oceanus and Thetys). Her Roman equivalent was Salacia, god Neptune's wife.
Her name means "The third one who encircles (everything)" (because the ancient Greeks thought the whole world was encircled by a river-god, Oceanus).
She spent most of the time singing and dancing with her sisters. One day Poseidon, god of the sea, saw her near the island of Naxos and fell in love with her. He kidnapped her and made her his wife.
In another story, Amphitrite, being a shy girl, fled away and hid somewhere in the Atlantic Ocean. Poseidon wanted desperately to find her, so he sent all the marine creatures to look for her. Delphinus, a dolphin, managed to find her and convinced her to go back to Poseidon and marry him. That's what made Amphitrite goddess of the sea. (By the way, the dolphin who found her was turned into a constellation by the grateful Poseidon).
Amphitrite goddess of the sea has no special attributions (but sometimes she can calm the waters), she is just the mistress of the sea (and sometimes she is considered the mother of the marine creatures, but they existed even before she became queen of the sea). Some even say that she is just a personification of the sea.
She also appeared briefly in the story of Theseus. King Minos asked the hero to prove that he really was Poseidon's son. A ring was thrown into the sea and Theseus had to bring it back. The sea creatures found the ring and gave it to the hero, while Amphitrite goddess of the sea received him in her underwater palace and gave him a golden crown, which he took to Ariadne as a wedding gift. (As you can see, Amphitrite was the ideal wife, as she was never jealous of her husband's love affairs.)
Amphitrite and Poseidon had a son, Triton (a fish-man or a he-mermaid) and a daughter, Rhode, and maybe another daughter, Benthesicyme.
In art, she was usually represented with her husband, together with many real and imaginary sea creatures, as you can see in the following pictures of Amphitrite goddess of the sea.
Triumph of Neptune and Amphitrite, by Nicolas Poussin