WASHINGTON—President Donald Trump ousted his communications director after only 10 days and introduced his new chief of staff, a former Marine Corps general who has the task of imposing more discipline in the West Wing, following one of the most turbulent weeks of the administration.

Anthony Scaramucci was removed from the communications director post on Monday, becoming the seventh major administration official to leave in Mr. Trump’s first six months. Mr. Scaramucci was ousted at the urging of the new chief of staff, retired...