A state is a relatively well-defined unit, based upon the largest unified government that maintains strong de facto power over a region. States are generally characterized also by some degree of recognition in the international community, and reasonably well-defined borders. A region could fail to be a state either because it is part of some other state, or because it is so chaotic that no institution holds de facto power over the region. Most of what we think of as "countries" are unambiguously states--somewhat ironically, the United States is very much one single state at this point.
A nation is a much more fraught concept; one person's "nation" is another's "ethnic group." The idea is that people of a nation share some sense of cultural unity; they speak the same language, they consider their heritage shared, they have cultural features in common that are distinct from others. Yet in practice, these boundaries are often extremely fuzzy---is Germany a nation, or just Bavaria? Is Canada a nation, or specifically Quebec? Could Californians be considered a different nation from Texans? What about Southerners versus Yankees (there was a civil war over that, after all)?
Still, it does seem to be the case that a single state can encompass multiple nations. The United Kingdom is the classic example; Scotland and Wales really do seem to be distinct nations from England, but all are part of the same state, the United Kingdom. When the British Empire was stronger, it was even sensible to say that Canada, Australia, and India were part of the same UK state, but India especially was clearly not the same nation.
Perhaps the best definition of a nation is the maximum unit of strong solidarity (as opposed to de facto power). Your nation is the people you think of as "like you", as "on your team." But precisely because people vary in their concepts of solidarity, this definition doesn't result in clearly-defined national boundaries.
This is a feature, not a bug; it shows that our definition is capturing the real ambiguity. In the real world, the ambiguous and fraught defining lines between nations have been a source of much conflict. People who think of themselves as separate nations often seek to form separate states, while people who do not think of those people as part of a separate nation will often try to maintain unity against that separation. And as national identities change, the desired boundaries change as well--often violently.