There are many different types of government in the world, including full presidential systems, like the U.S.; semi-presidential systems that have both a president and a prime minister, like the Central African Republic; parliamentary republics that have only a prime minister, like in Iraq; constitutional monarchies that have both a prime minster and a monarch but grant the monarch limited power through the constitution, such as in the UK; absolute monarchies in which the monarch's have complete rule that is not controlled by and constitution, like Saudi Arabia; theocracies in which the head of state is chosen by a religious hierarchy, like in Vatican City; military states in which the military has control over the government and even political executives are members of the military, like in Thailand; countries with single political powers, which are commonly communist countries like China and Cuba; and many, many other forms of government ("List of countries by system of government"). There are also many ways in which all are similar and different. As we are limited in space, below are a few ideas to help get you started.
The presidential and parliamentary forms of government are the most similar of the list in that both forms of government have a head of state. They are also similar in that both are democratic forms of government. However, they differ drastically in terms of power and the election process. In a presidential state, candidates for the presidency are chosen by the government's parties. On voting day, the citizens vote for electors in the Electoral College to vote for the candidate of the citizen's choice. Sometimes citizens vote for a particular president because they feel the president's party should be the leading party in office; other times citizens vote for who they think is the best individual person to lead the country ("How the President of the United States is Elected"). In contrast, in a parliamentary government, the prime minister, which is the head of state, is elected in to office when the party he/she represents is elected into office("Parliamentary Democracy"). In other words, unlike in the US, an individual person is not voted into office; instead, the most popular party is voted into office and the head of that party becomes prime minister.
Other than differences in the election process, there are also considerable differences between a president's power and a prime minister's power. For one thing, both a president and a prime minister have the authority to appoint Cabinet members; however, there is a difference in how the Cabinet is used. For a president, the Cabinet serves as a group of advisers. It is ultimately the president who makes the final decision, leaving the president with more power over the Cabinet ("The Executive Branch"). However, in a parliamentary government, once the prime minister appoints the Cabinet members, it is the Cabinet that makes executive decisions, which become government policy once supported by the House of Commons. However, like the US Cabinet, the Cabinet still has limited power, especially since the prime minister can at any time dismiss Cabinet members. If a government moves from being a Cabinet government to a prime ministerial government, as the UK government was accused of so doing under Tony Blair, the prime minister "drives the agenda of Cabinet meetings," leaving any points in which the Cabinet and the prime minister disagree out of the discussion ("The Cabinet and British Politics"). Also, unlike the US president, the prime minister is able to add new members to the House of Lords, which would be the equivalent of the president adding new members to the Senate or House of representatives, something of course a president cannot do. Since a prime minister can have the ability to drive the policy-making agenda, ultimately a prime minister has more policy-making power than a president.
A prime minister also has more additional authority to appoint leaders than a US president does. Both a president and a prime minister have the authority to appoint ambassadors and judges, but a Prime Minister has the authority to appoint all top civil servants and even bishops, while a president can only also appoint Cabinet secretaries and other Cabinet positions ("President versus Prime Minister").