I assume that this question is asking about the United States. In the United States, it is the legislative branch of government (the Congress) that has the right to impeach officials of the federal government.
In the US, we have separation of powers. We want the various branches of government to acts as checks on one another. Therefore, we would not allow the executive branch to impeach the president since he (or someday she) is the head of that branch. Instead, we have given that power to the Congress since it is the branch that is most directly elected by the people.
In our system, only the House of Representatives may vote to impeach a president, Supreme Court justice, or other official. Only the Senate has the right to try an official who has been impeached. Since the House of Representatives and the Senate are the two houses of Congress (which is the legislative branch), we can say that the legislative branch has the power of impeachment.
The House of Representatives has the power of impeachment, while the Senate has to power to try those officials who have been impeached.
In America, only two presidents have been impeached: Andrew Johnson (in 1868) and Bill Clinton (in 1998). However, it should be noted that neither were convicted by the Senate. For his part, Andrew Johnson was impeached for illegally removing Secretary of War, Edwin M. Stanton, from office. Johnson essentially violated the Tenure of Office Act by doing so.
Edwin Stanton had long been a thorn in Johnson's side; he was a radical Republican who opposed Johnson's lenient reconstruction policies. Basically, Johnson's easy stance led to Confederate loyalists returning to their former positions of power. As a result of this amnesty, new Southern governments were able to institute the Black codes, leading to the preservation of slavery in all but name only. Read more about what led to Andrew Johnson's impeachment here.
In President Clinton's case, he was impeached by the House for committing perjury regarding his extra-marital affair with Monica Lewinsky. Read more about it here.
For its part, the House of Representatives has initiated more than 60 impeachment proceedings. However, less than a third has led to full impeachments. To date, fifteen federal judges have been impeached, but only 8 have been removed from office by the Senate.
A president who was almost impeached was Richard Nixon. He resigned before he could be impeached and convicted for the 1972 break-in at Democratic Party headquarters in Washington D.C. Read more about it here.
In article II, section 4 of the U.S. Constitution, the power to impeach is given to the House of Representatives. The Senate serves as the court for impeachment trials. The House of Representatives and Senate make up the legislative branch of the U.S. Government.