Impeachment of a president. What is it really? It makes no sense to me the way it is used! Help explain?

the Constitution says "The President, Vice President and all civil Officers of the United States, shall be removed from Office on Impeachment for, and Conviction of, Treason, Bribery, or other high Crimes and Misdemeanors." As you know everyone says that president Clinton was impeached, however, was he really? To be impeached requires a simple majority. To be convicted of impeachment requires a 2/3 majority at a trial by the US Senate! Clinton was not convicted by the senate, nor has any other president ever been convicted. How can you have one, without the other. They do get a trial!