How the systems of government in sparta and anathens influenced western political thought?

Well, as my teacher happily likes to inform my class, Spartan politics essentially created the idea of a Constitutional Monarchy. A Constitutional Monarchy is basically what we have in the United Kingdom, where the King/Queen is a figure-head and doesn't actually run the country.

Athens however, was a Direct Democracy (due to the fact that they had slaves doing all the work and allowing the citizens the leisure time to debate politics) and is one of the two main forms of Democracy we have that exists today.

I don't know of any country or nation which employs Direct Democracy, normally it's Representative Democracy (voting in MPs, Senators, to do the voting for the common people) but I do believe it's influenced Western politics because, without the Athenians developing Direct Democracy in the first place, we might not have had Democracy as we know it today. This is because, most of the Greek city-states, after ridding themselves of their own monarchies, normally ended up with Tyrannical rule (not the same as it means nowadays) or Oligarchy's (which is basically the total rule of a country/nation by a select few; like Elders, Military Councils etc.