Extinction was the end of a species caused by either the death of the entire population or over a long period of time in which the species was slowly reduced in number until the last members eventually died.
Major catastrophes could cause hundreds to millions of species to die out all at once. In cases of planetary destruction, all the species native to the planet could be driven to extinction (unless any native species had access to space travel).
In 1981, the arrival of the Master on Logopolis fatally disrupted life there, and he killed Logopolitans at random with his Tissue Compression Eliminator. Eventually, because their Block Transfer Computations no longer worked, the planet itself began to crumble and lose coherence, as did the inhabitants. The entropy field caused by the destruction of Logopolis also destroyed a portion of the universe, such as the planet of Traken. (TV: Logopolis)
At the climax of the century-long war between the Thraals and the Olabrian, the Olabrians destroyed the Thraal's homeworld, wiping out all life on the planet. (PROSE: Ship of Fools)
The Veil became near-extinct when their planet's star was extinguished. The last member of the race, Androvax, went on a vengeful vendetta, destroying twelve planets before finally being arrested by the Judoon when he attempted to destroy Earth. (TV: Prisoner of the Judoon)
Survivors of attacks supported by the Etydions banded together to vaporise their homeworld, cleansing it of all life. One Etydion survived, arrived on Earth and tried to convert the human race to his kind, but when he realised what he was doing he reversed the process and died. (TV: The Custodians)
The Time Lords erased the Charons before they existed as they feared that all other life in the Universe could not co-exist alongside them due to their ability to warp the fabric of the universe. Only one Charon somehow survived the death of its people and constructed a pocket universe. The Seventh Doctor confronted the Charon and learned that, after being alone for so long, it wanted to die. The Doctor sadly granted its wish. (PROSE: Sky Pirates!)
The Klempari defence was a common ruse in interrogation. The subject would claim to be the last of their kind. When the Eleventh Doctor questioned the Silurian, Alaya, she claimed to be the last of her species. He knew she was lying because he was the last of his species, and told her not to insult him. (TV: The Hungry Earth)