Princess Neptuna is a royal princess, that is likely the daughter of King Neptune and Queen Amphitrite, which she was named after apparently. She has never made a physical appearance, but was the focal point of SpongeBob and the Princess, which is a book that came out around the time of Season 3. She does not appear as a Mermaid, but a tan colored fish, with a pink dress, and a golden crown, accompanied by a fancy, purple snail, on a diamond leash. She signs a picture for Squidward, and gives a nice comment on SpongeBob's appearance.
Most Bikini Bottomites think of her as famous by rolling out the red carpet, they are not even sure if she was coming.
Unlike the rest of the Neptune family, she's a fish for some unknown reason and not a mermaid. It's possible that she was adopted, or she is not part of the family and is in a different royal family.
She and her supposed sister Mindy are the only two not named after gods and goddesses of mythology, while her supposed father, Neptune, is the Roman god of the seas, Amphitrite is the goddess of the seas, and Triton is the son of Neptune in mythology.
Her name is similar to Neptune, the only difference being that her name ends with an "A," not an "E."