So what exactly is "Smart Power"?

Hillary Clinton used a new phrase in her speech to the Senate Foreign Relations Committee (currently headed by former Presidential Candidate John Kerry):

"The President-Elect and I believe that foreign policy must be based on a marriage of principles and pragmatism, not rigid ideology. On facts and evidence, not emotion or prejudice. Our security, our vitality, and our ability to lead in today’s world oblige us to recognize the overwhelming fact of our interdependence.
I believe that American leadership has been wanting, but is still wanted. We must use what has been called “smart power,” the full range of tools at our disposal -- diplomatic, economic, military, political, legal, and cultural -- picking the right tool, or combination of tools, for each situation. With smart power, diplomacy will be the vanguard of foreign policy. This is not a radical idea. The ancient Roman poet Terence, who was born a slave and rose to become one of the great voices of his time, declared that “in every endeavor, the seemly course for wise men is to try persuasion first.” The same truth binds wise women as well."

[Transcript, New York Observer, January 13, 2009)

In international relations, the idea of "soft power" refers to the ability of a political body to indirectly influence decisions made by other political bodies (states etc.) - by cultural or ideological means. Whether or not a states is able to exercise this power depends on reputation and "credibility" (Joseph S. Nye - Propaganda Isn't the Way: Soft Power).
Hard power is probably best described by giving the example of Russian gas policy and how it is used to influence political outcomes in countries dependent on Russian gas. Hard power refers to political and economic means to outcome the decisions of political bodies. It is usually described as aggressive and threatening. In contemporary interpretation, an actor who has to resign to hard power to get his way is probably not as well-off as one who can rely on soft power - but probably that will change in the coming years?
"Smart power" is a term coined as a reaction to the US strategy in Iraq and refers to the combination of hard power and soft power to a "winning strategy" (Nye: In Mideast, the goal is 'smart power'). Clinton used it as a promise: during the past eight years, the State Department has lost its influence as the Defense Department exercised more and more influence on foreign policy.
When I wrote about Obama's and McCain's positions on foreign policy expressed in Foreign Affairs, I already mentioned that there are some differences, but possibly not quite as many as one would expect from two candidates who seemed so diagonally opposed during the campaign. So, what about the United Nations?

"We should also use the United Nations and other international institutions whenever appropriate and possible. Both Democratic and Republican presidents have understood for decades that these institutions, when they work well, enhance our influence. And when they don’t work well – as in the cases of Darfur and the farce of Sudan’s election to the former UN Commission on Human Rights, for example – we should work with likeminded friends to make sure that these institutions reflect the values that motivated their creation in the first place."

"Likeminded friends". That sounds nicer than anything previously said about the "coalition of the willing", but we will see how this idea plays out in the future. The common thread and most easily identifiable idea both in Obama's positions and Clinton's speech is the "renewal of American leadership" - the idea that in the past eight years, the US has not fulfilled its promise as the City Upon the Hill, a symbol apparantly nobody can question or do without. In her speech, Clinton calls this a "promise to humanity" - and this particular occasion a "new American moment".

Guardian: Hillary Clinton backs 'smart power' to assert US influence around world, January 13, 2009
Politico: Clinton endures light grilling, January 13, 2009