In the original Shang-Chi comics from the 1970s, the Ten Rings have a bit of a wacky origin story. According to the official Marvel website, the rings come from the planet Maklu-IV, which is home to the “dragon-like race Axonn-Karr (or Makluans).” These rings, which were originally small enough to fit on a person’s finger (instead of their arm, like in the film), were used to power the Makluans’ spaceships.
Which is exactly how they ended up on Earth. Mandarin, the villain from Iron Man 3 and Shang-Chi’s father, discovered the rings on a crashed spaceship, likely from Fin Fang Foom, the comics’ most-recognized Makluan.
In the comics, each ring held a different power: Ice Blast, Mento-Intensifier, Electro-Blast, Flame Blast, White Light, Black Light, Disintegration Beam, Vortex Beam, Impact Beam, and Matter Rearranger.
In the comics (and in Iron Man 3), Mandarin learns to utilize the rings for his own power, which leads to his villain origin story. Fin Fang Foom is not in Shang-Chi and the Legend of the Ten Rings, but that doesn’t mean we won’t see him in later Marvel installments.
It’s assumed that Shang-Chi inherited the rings after defeating his father.