The Master Race was created when the Master stepped into the Immortality Gate and transferred his genetic template to the more than 6 billion human inhabitants of the planet Earth in 2010. (DW: The End of Time)
History
When the Master stepped into the Immortality Gate, an alien artifact being reverse-engineered by billionaire Joshua Naismith, it sent out his genetic structure to nearly every human on Earth, turning everyone into exact copies of him. At least two individuals escaped this transformation: Donna Noble (who appears to have been protected due to the influence of her DoctorDonna transformation), and Wilfred Mott, who was protected by special radiation shielding at the time.
With all humans, including US President Barack Obama and the leaders of UNIT, now essentially the same man, The Master ordered his self-proclaimed Master Race to take control of the planet (with the original Master as their undisputed leader). After the Doctor's escape, the planet's military resources were dedicated to searching for him.
Once the human race had all become The Master, they all heard the incessant drumming that had driven the Time Lord mad over the years. Soon after, an the Master realised he could use the combined mental power of the Master Race to send a signal of the drumming through time and space and help the Time Lords return to the universe. They successfully made contact, and a White-Point Star diamond was sent to Earth, which The Master Race recovered and used as a power source for the Gate to pull Gallifrey out of the Time War. Soon, Rassilon and other Time Lords began to arrive on Earth through the Immortality Gate, though the Master's efforts also caused Gallifrey itself to manifest near Earth, threatening the planet with destruction.
One of Rassilon's first acts after arriving on Earth was to reverse the Master's transfer of genetic code, which resulted in the Human race returning to their original form.
Non-human entities such as Sarah Jane Smith's computer Mr. Smith did record the events surrounding the transformation. The Doctor and the alien scavengers Addams and Rossiter were also unaffected. Any other non-human species present on Earth at the time (e.g. Bayldon Copper) were likewise not affected. It also seems that Luke Smith was unaffected, as he remembers seeing everyone with a "different face", and he is not a normal human being.
Biology
As the Master Race, every human on Earth was rendered identical in every way to the Master with no apparent gender or age differences. It is unclear whether this temporarily turned the human race also into Time Lords. Also unclear is whether the change applied to infants and newborns, or whether, had the Master Race been allowed to continue, any possibility of procreation existed. Wilfred Mott also wondered if dead bodies were also affected. The Doctors reaction hints they were.
It is also never made clear whether or not they posessed the superhuman abilities that the Master had at the time due to his unstable form. They likely did not, or they could have hunted Donna Noble much more quickly, and the guards could have flown to the crater containing the white point star.