I can't comment on the evolution of the terms over time, but today, academics generaly distinguish them as follows:

A state refers to the government that administers a certain territory. A more formal definition, which is still widely used today, was developed by Max Weber in the 19th century, where he defined a state as an entity which has a monopoly on the legitimate use of force within a specific territory.

A nation is much more vague, but it generally refers to a distinct ethnic, cultural, or linguistic group that inhabits a certain region. In other words, it's a cultural group rather than a formal political one. This is why the term "nationalism" is generally associated with people who identify strongly with a specific ethnic or cultural group.

These two terms are where the term "nation-state" comes from; this term refers to a group of people with some sort of common culture who live under a government that controls the territory in which they reside.

The term country generally refers to the land itself. I.e. not the government (the state) or the people (the nation), but the physical geographical place.

Academics - historians, political scientists, etc. - are really the main group of people that abide by these distinctions. In everyday language, the three terms are often used interchangeably. In the US, the distinction becomes even more confounded because "state" generally refers to one of the 50 states, which by the formal definition coined by Weber aren't actually states because they are not autonomous and have to answer to the central government in Washington.