You Do Need power plug adapter while traveling from Réunion to Nauru.

Here is Why You Need An Adapter

The outlet and plug type used in Nauru is I and in Réunion are C and E. This is how they look like.

Réunion

In Réunion, outlets of type C and E are in use.

Type C

Type E

Nauru

In Nauru, outlets of type I are in use.

Type I