Is it possible for a United States Secretary to be removed by the House of Representatives or the Senate?
Not directly, as:
Members of the Cabinet serve at the pleasure of the President, which means that the President may dismiss them or reappoint them (to other posts) at will.
Nonetheless can Congress get rid of a criminal secretary via its Congressional oversight powers and by using the Impeachment procedure (cf Article 2, Section 4 of the U.S. Constitution), but only if they are found guilty of some sort of crime.
Impeachment is a two-step process. First the House of Representatives impeaches the official and only after that does the Senate try him. If found guilty by the Senate the official is immediately removed from his duties.
Yes, but not just because they feel like it. Cabinet secretaries are civil officers, so they fall under impeachment rules. Article 2, Section 4 of the U.S. 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.
All impeachments require both Congressional houses; the House of Representatives impeaches the official, and the Senate tries them. I think historically it's only happened once to a secretary; William Belknap, Grant's Secretary of War, was impeached in 1876.
The "Separation of Powers" principle of the Constitution makes Cabinet Secretaries of the Executive branch answerable to the President, not to Congress.
Nevertheless, Congress has a very limited Constitutional right to remove these members of the Executive Branch through the same mechanism as they can remove the President himself; through "impeachment" for "high crimes and misdemeanors."