The argument against impeachment in the Clinton case is that he perjured himself in a civil matter that was not connected to the office of the presidency. Most scholars, but not all, interpret the impeachment clause of Article II Section 4 to be limited to crimes committed in capacity as president, in other words using the power of the presidency to engage in criminal activity (as Nixon did).
Article II Section 4: 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, and other high Crimes and Misdemeanors.
This interpretation rests largely on the framers' decision to only specifically identify Treason and Bribery, and use the word 'high' in front of Crimes and Misdemeanors. The framers clearly understood that men who became president, especially in their day, came from prestigous backgrounds. They often were very wealthy, owned a lot of land and business. They were men of high ambition. And it may be that they would get elected to office with outstanding litigation against them or their businesses. They may be accused of all sorts of things in their capacity as private citizens, but being accused and proven to be an embezzeller or perjurer in a man's private life would disqualify that man from holding high office.
There is a lot of gray area here. If we elected someone in 2008 only to find out that in 1998 he murdered someone, should he be impeached?? It is difficult to imagine that the country could tolerate an accused murderer in the office for 3 more years. Fortunately, this hypothetical has not been tested.