Canadian folkie and rocker Neil Young would not be eligible to run for President given his birthplace. But in 2006, he wanted to remove one (George W. Bush).
Impeachment is the legal way to turf a president, between elections. It is for Presidents (and state officials) alleged to have committed high crimes (like misuse of office and abuse of authority, treason and bribery).
The process is thus:
The legislative branch of the USA, the House of Representatives, must agree with a simple majority (50% plus one) of its members to impeach.
Then, the other legislative branch, the Senate, holds a court of sorts, chaired by the Chief Justice of the US Supreme Court. This body then deliberates on whether to make the impeachment official. This Senate trial requires a two-thirds majority of its members to make such a determination.
President Andrew Johnson narrowly won the Senate vote by one vote in 1868 to avoid impeachment. President Bill Clinton was acquitted of impeachment by the Senate in 1999. However, both were impeached, at least at the House of Representatives stage. They were impeached, but not removed from office.