The Nauru Reed-warbler Acrocephalus rehsei (Theme Birds on Stamps) (p. 78 of ResearchGate) is often considered distinct from the Nightingale Reed-warbler Acrocephalus luscinius (ARKive) and is found nowhere else.
A subspecies of skink Emoia
arnoensis nauru (BHL)
is endemic and a recently collected skink may prove to be a new species
(p. 68 of ResearchGate).
Insects known solely from Nauru include an tidal reef bug Corallocoris (or Omania) nauruensis (Discover Life) (Internet Archive) and an undescribed leafminer moth of the genus Stigmella (p. 42 of ResearchGate). Endemic land snails include Trochomorpha insolata and Sturanya subsuturalis (both on p. 43 of ResearchGate). A barnacle Balanodytes echinoideus (WoRMS) is also known only from Nauru.
Although it has been claimed that a species of Phyllanthus might be endemic, according to "The Flora of Nauru" (Smithsonian 9 MB pdf file), there are no endemic vascular plants and the claimed endemic turned out to belong to the widespread Phyllanthus societatis (p. 21 of environnement.pf pdf file).
For an overview of Nauru's biodiversity see (ResearchGate). Nauru is part of the Eastern Micronesia Tropical Moist Forests ecoregion (EoE).