According to what I've read: the House has the ability to impeach a President, however this doesn't mean the President has to leave office. He only has to leave office if the Senate finds him guilty. Thus, impeachment is just a formal accusation or charge, and the Senate is the judge/jury in which the President is either found guilty or not-guilty. A positive or negative result in the Senate doesn't change the fact that the President was impeached.
The House of Representatives shall chuse their Speaker and other Officers; and shall have the sole Power of Impeachment. (1)
Removal from office if convicted, is separate from the act of impeachment itself. (2)
If a simple majority of the those present and voting in the House approve an article of impeachment, then the president is impeached. (3)
Is this the correct way of thinking? Because lately, in the news, and conversationally people seem to think that he isn't impeached until after the Senate renders their verdict, and some argue that the President isn't impeached until the articles are sent to the Senate (answered here). The entire system is even called the "impeachment process" which seems to indicate multiple "steps" are required for impeachment (rather than just the House vote).
- US Constitution: Article I, Section 2, Paragraph 5