Impeachment and removal
Impeachment and removal are the two steps in taking a high government official, such as a president or a judge, from his position. It is typically used in cases of malfeasance (see also recall election).
The first step, impeachment, is a formal accusation by a simple majority of the House of Representatives. This vote leaves the accused in office, while he is "tried" by the Senate. A two-thirds majority there results in removal from office.
Sixteen federal officials have been impeached (i.e., charged) by the House of Representatives, but the Senate has removed (i.e., convicted) only seven. 
The United States House of Representatives has "sole power of impeachment." The House has to vote on articles of impeachment. If they pass by a majority, the President is not yet removed from office, until (and unless) the next step is completed.
The impeachment case is sent to the United States Senate for trial. If the Senate votes by a two-thirds majority to convict, then the person will be removed from office.
Impeachment in 19th Century America
By 1804, Jeffersonian Democratic-Republicans had control of Congress and set their sights on removing Federalist Supreme Court Justice Samuel Chase. The House of Representatives impeached him, and trial began with the Senate as the jury. His prosecutor was John Randolph, a partisan Jeffersonian who was distantly related to the Federalist Chief Justice John Marshall. 25 out of 34 Senators were Jeffersonians, so conviction and removal of the Federalist Chase for his political biases seemed likely.
But the effort failed. During the trial Randolph himself criticized Jefferson over a separate issue, the infamous Yazoo land fraud in Georgia, an issue that split Jeffersonians. Meanwhile, the case against Chase amounted to little more than some intemperate remarks he made while sitting as trial judge in a grand jury proceeding. The Senators were unimpressed, and at most 19/34 voted to convict on any of the charges, far short of the requisite 2/3rd. The independence of the judiciary was established by this failed effort to remove a Supreme Court justice. However, the episode did encourage future justices not to engage directly in politics. No effort to impeach a Supreme Court justice has since made any progress.
After the Civil War, President Andrew Johnson became the enemy of Radical Republicans who controlled Congress. President Johnson vetoed their legislation and even called their leaders "traitors". Flush from winning the war, the likes of Republican Senators Thaddeus Stevens (PA) and Charles Sumner (MA) were not about to back down. Sumner, after all, was the fellow who once delivered an unconscionably disrespectful speech against elderly South Carolina Senator Andrew Butler, whereupon his outraged nephew Preston Brooks beat Sumner senseless with a cane.
When the House of Representatives impeached President Johnson by an overwhelming vote of 126 to 47, his ouster appeared to be a fait accompli. President Johnson had violated the Tenure in Office Act by dismissing his Secretary of War, and he had been intemperate in his name-calling of key senators. In addressing the jury of 54 senators, the prosecutor referred to President Johnson as an "accidental Chief" and "the elect of an assassin." Witnesses testified for both the prosecution and the defense.
The prosecutors – called “managers” in impeachment trials – were confident of victory. Manager Thaddeus Stevens described President Johnson as the "wretched man, standing at bay, surrounded by a cordon of living men, each with the axe of an executioner uplifted for his just punishment." Manager John Bingham brought the public galleries to their feet with his oratory: "May God forbid that the future historian shall record of this day's proceedings, that by reason of the failure of the legislative power of the people to triumph over the usurpations of an apostate President, the fabric of American empire fell and perished from the earth."
2/3rd vote was necessary for conviction, and it all turned on Senator Edmund Ross of Kansas. Like real jurors, he spoke to no one during the proceedings and no one knew which was he was leaning. But he voted "not guilty," and President Johnson was acquitted by one vote. The Radical Republicans were defeated, and never again enjoyed their unprecedented power.
Impeachment in recent history
In 1974, no impeachment proceedings had occurred since the Johnson trial over a hundred years earlier. Most Americans thought of impeachment as something quaint and bizarre in the history books.
The House of Representatives decided to begin impeachment proceedings against Richard M. Nixon, on charges of perjury and obstruction of justice in connection with the Watergate affair. At the time, congressmen and their staffs were unfamiliar with the law and proceedings involved with impeachment, and needed to research the question.
On July 27, 1974 the Committee on the Judiciary voted to recommend three articles of impeachment. The articles concerned his obstruction of justice during the Watergate investigations. Before his near-certain impeachment and conviction could occur, Nixon resigned from the presidency on August 9, 1974, after key Congressional allies told him his fate was all but assured. Thus, Nixon was never actually impeached.
President Bill Clinton was impeached by the House of Representatives on December 19, 1998, on charges of perjury and obstruction of justice relating to testimony he had given in a civil lawsuit a former Arkansas government employee had brought for sexual harassment. He was acquitted by the Senate on February 12, 1999 on a party-line vote.
President Clinton was doing well in the polls and ready to pick his successor for the White House by late 1998. His leadership enabled the Democratic Party to do surprisingly well in the mid-term elections, and President Clinton was seeking to leave a lasting legacy.
The House of Representatives impeached him for making false statements under oath, which is perjury, and trial was scheduled for the Senate in 1999. First Lady Hillary Clinton accused her opponents of being part of a "vast right-wing conspiracy" against her husband. The Democratic Senators all lined up behind President Clinton, promising to vote for acquittal.
Republicans enjoyed a majority in the Senate but lacked votes anywhere close to the requisite 2/3rd majority to convict. President Clinton was a lame-duck at that point anyway, and the real battle was over his ability to influence politics beyond his term.
The managers of the impeachment presented their case, but the Senate prevented them from calling key witnesses such as Clinton’s closest aides. The procedural rules hamstrung the trial and served to protect the president. Without the presentation of the full case, President Clinton easily survived the final vote.
But the victory was Pyrrhic. The Democratic presidential nominee in 2000, Al Gore, felt he had to distance himself from Clinton and that probably made the difference on Election Day. Since then, Clinton has had little success in campaigning for candidates, failing with Governor Gray Davis in California in the recall. Hillary Clinton, pollsters choice to run in 2004, apparently felt it was too soon to make an attempt. The political ramifications of the impeachment trial continue.
In colloquial English, speakers frequently fail to distinguish between the accusation by the House, and the (possible) subsequent removal by the Senate. Thus when someone says, "That official was impeached," you may not know whether they meant charged but not convicted or removed from office.
Two US presidents have been impeached but neither was removed from office.