For the past two years, the United States has been lead by one political party (Democrats). When all three elected portions of our government (Senate, House and White House) are in the hands of one political party, the government seems to either move in an extreme liberal or conservative direction.
The pros of a divided government require a moderate position on major issues or nothing in Washington D.C. gets done. Another big thing on the pros side to divided government is that political extremism is kept in check. For much of US history, the American people have made a choice to not leave one political party in power too long in both the Congress and the White House.
On the cons side of divided government, there is that nagging problem of some things needing to get done in Washington, but no moderate position can be found to bring the two political party's together. Most Americans when asked by pollsters say that they want politicians in Washington to get along and work together to solve problems. However, that wish seldom becomes a reality - because both the Republican and Democratic Party's have been taken over by extremist who do not want to compromise on anything.