Give them legs. They could be separate from birth and just covered with a swimsuit. Or they could be joined at birth and separate during puberty. These follow somewhat from some...
Real life precedents
Tiffany Yorks, Shiloh Pepin, and Milagros Cerrón were born with a rare condition called sirenomelia or "mermaid syndrome", where the legs did not separate in the womb. Though Shiloh could not undergo separation surgery because her legs' blood vessels were too tangled, Tiffany and Milagros had their legs successfully separated. Both needed extensive rehabilitation to walk, and Tiffany's continuing mobility issues echoed the admonition of the Sea Witch in H. C. Andersen's story that "at every step you take it will feel as if you were treading upon sharp knives."
A monofin is a paddle that clips to a swimmer's feet, giving more surface area for a dolphin kick. This leads to improved performance in swimming and free diving. Cloth covers for the legs and monofin allow professional swimmers to perform as mermaids in public. These covers are also sold commercially by various firms; others are custom-made.
The Moken people and Sama people are two Austronesian ethnicities living seaborne nomadic lifestyles. These "Sea Gypsies" are known for seeing underwater and holding their breath for several minutes.
Michael Phelps won a dozen Olympic golden medals for swimming in 2008 and 2012. In his autobiography No Limits, he attributes this to his long, thin torso, long arms, short legs, and large feet connected to hypermobile ankles. Pauli Poisuo of Cracked called him "basically a seal." Similarly, Icelandic fisherman Gudlaugur Fridthorsson swam for six hours in freezing water in 1984. After his full recovery, it was discovered that his body fat resembled that of a seal more than a human.
Francine DeGrood Taylor's answer that they are descended from humans who adapted to a semiaquatic lifestyle sounds like it's on the right track. But there are two ways this could go, even without any sort of magic.
One way leaning toward hard science is to make merpeople just a human ethnicity. This works well for "selkies," who can change between human and seal-like forms by putting on a sealskin wetsuit. Ethnic merpeople may resemble Sea Gypsies, with physiologic traits resembling those of Phelps and Gudlaugur, and wearing bamboo-tech monofins with a covering to hold the legs together and reduce drag.
Another more extreme adaptation might be to have babies born with webbed legs, similar to sirenomelia. The children prefer to move through water, not straying far from the beach or the river, because moving on land requires placing one arm and the heel forward and then pulling the butt up to meet them. (Shiloh demonstrated this technique in one of the TV documentaries about her.) Around puberty, hormonal changes cause the webbing to retract, separating into legs over the course of the next few years. Puberty would be a pain, as adolescents would need to learn how to walk from scratch.