What name is given to those species: (a) that are on the verge of vanishing from earth? (b) that have died out completely.

Species on the verge of vanishing from Earth are called endangered species. These are organisms whose population numbers have declined to critically low levels, putting them at extremely high risk of extinction in the near future if protective measures are not implemented. The International Union for Conservation of Nature (IUCN) maintains detailed classifications, with "endangered" representing species facing a very high risk of extinction in the wild, while "critically endangered" describes those at even greater immediate risk.

Species that have died out completely are termed extinct species. Extinction means no living individuals of that species exist anywhere on Earth, representing an irreversible loss of biodiversity. Examples include the dodo bird, which became extinct in the late 17th century, and dinosaurs, which disappeared approximately 65 million years ago. Once extinction occurs, that unique genetic lineage and its ecological role are permanently lost, which is why conservation efforts focus heavily on preventing species from reaching this final, irreversible stage.