In most countries that have a royal family, there is also a prime minister who leads the executive branch of their government. They usually have a parliamentary system. The prime ministers have a lot of power that is similar to a president. The prime minister may appoint, hire and fire members of the cabinet. The duties of the prime minister may also help pass the bills into law in the legislative branch.
Since the United States has a democratic republican government, we do not have a prime minister. Instead, we have a president. The president and the prime minister have a lot of similarities except the president probably has a little more power than the prime minister since the prime minister must also consult the royal family.