Wikipedia lists 43 countries with a presidential system of government where the President is the Head of State and Government.
In addition, there are 15 countries with a President that is a Head of State and Head of Government, but also has a position of Prime Minister that typically holds an advisory role to the President.
2 Countries have a Presidential system where the President is head of State only and the Prime Minister is head of Government, but is answerable to the President, not Parliament.
A large number of subnational level governments are Presidential including all 50 US states and its territories, and many if not most of Japan's 47 prefectures, despite the national government being a Constitutional Monarchy.
Wikipedia lists a Presidential system as having four catagories:
- Veto of Legislation
- Fixed term of office, usually with a lack of "No Confidence Voting".
- Appointment of Cabinet and Judiciary, with the ability to dismiss the former but not the latter.
- The ability to Pardon or Commute criminal sentences.
It should be pointed out that Iran is one of the 43 Nations with a full presidential style government but it wouldn't be fair to say it's of US style.
The article lists several pros and cons of a Presidential system, among which, the most important is the Stability of the government, since a President cannot be dismissed as easily as a Prime Minister. Having said that, as envisioned by the Founding Fathers of the United States, the Legislature, not the Executive, was to be the most powerful branch of the U.S. Government.