when a particular type of animal or plant stops existing:
species in danger of extinction
on the verge/edge/brink of extinction (=nearly extinct)
The breed was on the verge of extinction.
They were hunted almost to extinction.
Conservationists are trying to save the whale from extinction.
face extinction/be threatened with extinction
Many endangered species now face extinction.
when a type of person, custom, skill etc stops existing:
Their traditional way of life seems doomed to extinction.