What are examples of nations without a state?

Examples would include: Nigeria, and Liberia in Africa. There are nations (cultural groups) without States. Sometimes these cultural groups are scattered among several States and sometimes they are a minority in one State. Sometimes these stateless nations are unwelcome and are a cource of conflict.

Why is a nation not a state?

A nation, in the sense of a common ethnicity, may include a diaspora or refugees who live outside the nation state; some nations of this sense do not have a state where that ethnicity predominates. In a more general sense, a nation state is simply a large, politically sovereign country or administrative territory.