According to a U.S.C.C.B. press release: Cardinal Timothy Dolan of New York, president of the U.S. Conference of Catholic Bishops, congratulated President Barack Obama, Nov. 7, the day after his re-election as President of the United States. The U.S.C.C.B. released the text of the letter this morning:

Dear President Obama,

In my capacity as President of the United States Conference of Catholic Bishops, I write to express my congratulations on your re-election as President of the United States.  The people of our country have again entrusted you with a great responsibility.  The Catholic Bishops of the United States offer our prayers that God will give you strength and wisdom to meet the difficult challenges that face America.

In particular, we pray that you will exercise your office to pursue the common good, especially in care of the most vulnerable among us, including the unborn, the poor, and the immigrant.  We will continue to stand in defense of life, marriage, and our first, most cherished liberty, religious freedom.  We pray, too, that you will help restore a sense of civility to the public order, so our public conversations may be imbued with respect and charity toward everyone.

May God bless you and Vice President Biden as you prepare for your second term in service to our country and its citizens.
                                                                      
Sincerely yours,

Timothy Cardinal Dolan
Archbishop of New York
President
United States Conference of Catholic Bishops

Pope Benedict also sent congratulations [3] to President Obama (via telegram!) [4]. According to Vatican Radio:

In his Message, the Holy Father offers his best wishes to the President, and promises continued prayers on his behalf. The Pope assures the re-elected President that he will ask God to help him in his high responsibility to the country and the international community. The Pope also says he will pray that the ideals of freedom and justice, which guided the founders of the United States of America, might continue to shine through the nation as it makes its way in history.

The Director of the Press Office of the Holy See, Federico Lombardi, SJ, commented:

All hope that President Obama, confirmed in his office after the elections, will respond to the expectations of his fellow citizens; that he might serve right and justice for the benefit and growth of every person, in respect for those human and spiritual values, which are essential to the promotion of the culture of life and religious freedom, which are ever so precious in the tradition and culture of the American people, so that that people might be capable of finding the best ways to promote the material and spiritual welfare of all; so that it can effectively promote integral human development, justice and peace in the world.