There are two possible answers to this, depending on the context in which you are asking this question. 

First, if you are asking this in the context of the US political system, the difference is simply that a state is a political division of the nation as a whole.  We use the term “nation” to refer to the entire country.  The states are divisions into which the country has been broken so as to create a level of government that is closer to the people.

The second difference between these two terms is used in the broader study of political science.  When “state” is used in this way, it is another word for a country.  The state has a defined territory that belongs to it.  Within its borders the state is sovereign and no outside entity has the right to tell it what to do (without its consent).  By contrast, a “nation” is something like an ethnic group.  It is a group of people who share things like a common language and a common history. 

Some nations have states that belong to them.  One example of this is Japan, which is dominated by ethnic Japanese.  We would say that Japan is a nation-state.  Some nations do not have states.  An example of this is the Basque people whose native land is in Spain and France.  Some states are made up of more than one nation.  An example of this would be Belgium, which has Flemings and Walloons within its state boundaries.