Merfolk

Merfolk, (PROSE: Evolution, The Suns of Caresh) were a race of aquatic beings that lived in the seas of Earth. Female merfolk were known as mermaids, while males could be referred to as either mermen or merrows (PROSE: The Sorcerer's Apprentice)

Biology
Mermen had a fish's tails from the waist down. Their top half appeared human. They had a second set of inner eyelids. Mermen were unable to breath for long periods on land, (AUDIO: Cryptobiosis) however Magda, a mermaid rescued by Iris Wildthyme, lived on the surface for over thirty years without issue, albeit in a wheelchair for mobility. (PROSE: Enter Wildthyme)

History
King Neptune had mermaid servants. (COMIC: Guests of King Neptune)

Graham Stevens claimed that a mermaid rescued his grandfather from the Marie Celeste. (PROSE: Cat's Cradle: Witch Mark)

The Mermen attempted to stay unnoticed by the humans on the surface. Two Mermaids were captured in fishing nets and taken by Jacques de Requin to be sold.

Amfetriti, one of the two, died. Her daughter Galliteya, the other captive, was looked after by Peri Brown. Nereus led his people aboard the Lankester where they killed De Requin. Galliteya was returned to them. Nereus promised help for the crew who had jumped overboard. (AUDIO: Cryptobiosis)

In the 20th century Bea Nelson-Stanley and Edgar Nelson-Stanley saw mermaids leaping with dolphins. (PROSE: Eye of the Gorgon)

In the 1980s, Iris Wildthyme and MIAOW stopped Magda the Mermaid from destroying a nuclear power plant and then recruited Magda into MIAOW. (PROSE: Enter Wildthyme) In the Obverse, Magda was the Eighth Doctor's mother. (PROSE: The Blue Angel)

In the 30th century, the First Doctor and Ian Chesterton encountered several mermaids and mermen in the Circle Sea on Avalon. These mermaids were brought into existence on the planet through the nanobot system on the planet, which created the fantasies of 22nd century human colonists into real beings. (PROSE: The Sorcerer's Apprentice)

Alternate timelines
In an alternate timeline created by the Chronomasticon, Earth was taken over by mermaids. (PROSE: Playing for Time)

Myth
Mermen were also creatures of myth in Earth culture. Mermaids existed in the Land of Fiction, some whom were converted into Cybermen. (AUDIO: Legend of the Cybermen) Albert C. Richardson sarcastically asked if Barbara Wright was a mermaid that had lost her tail. (TV: The Chase) The Eleventh Doctor called an alien doctor who stalked the Fancy a mermaid as she had the appearance of siren. (TV: The Curse of the Black Spot) The Meron, a race native to Gorgol XIII, were also merfolk. (TV: Liberation)

Victorian showmen used to draw the crowds by taking the skull of a cat, glueing it to a fish and calling it a mermaid. The Ninth Doctor compared the Slitheen's Space Pig with this trick while explaining the Space Pig's true nature to Toshiko Sato. (TV: Aliens of London)

Behind the scenes
The Dax featured in the Decide Your Destiny book, The Corinthian Project, were compared to mermen, but given that their story had multiple endings, it's difficult to assert with certainty much about them.