More specifically, one should regard the advantages and disadvantages of unipolarity in regard to the exercise of military power. 

The United States, for example, outspends every other industrialized nation combined on the maintenance of its defense. This gives the United States an advantage that it has enjoyed since the Second World War: the ability to police conflicts in different parts of the world, particularly nations "threatened" by Communism or a totalitarian power. The US's gargantuan military power has given it the ability to ensure, or pursue, democracy throughout the world—even in nations that do not want it (e.g., Vietnam). 

The disadvantage is that other nations come to rely on the superpower for defense. This has been a subject of some controversy recently, as there are those who think that other industrialized nations have the capacity to build their own defense systems but have chosen to rely on the United States instead. The funneling of so much money into defense disallows the funding of other infrastructural programs. 

I assume that you are asking about the advantages and disadvantages of unipolarity for all the countries in the world.  

If this is the case, the major advantage of unipolarity is that it can be a very stable system.  In a unipolar system, there is only one superpower.  This country can act as something of a supervisor of the international system.  It can impose its will on the other countries, thereby maintaining stability.

On the other hand, however, a unipolar system can be somewhat oppressive.  The hegemonic power will be able to do more or less whatever it wants.  Lesser powers have no real way to resist it and are therefore left with no choice but to obey.  

In these ways, a unipolar system can be a tradeoff between stability and lack of freedom of action for the countries of the world.