This is a very hard question. When I think of best countries I think best to live in, best morality and best reputation. Now if your asking most influential or most powerful you'll get a totally different answer.
For me the best (Moral, Live in) countries are
#1. Japan - Japan is perhaps the most advanced country in the world. It lacks a hostile military it also takes care of its citizens (To my knowledge) The country also has more jobs then many other countries and medical advancements none others have. Also Democracy is at the countries core with people having many rights and freedom's. I would take being born in Japan to anywhere else in the world (Other then my country of Canada).
#2.The U.K - The U.K has been a symbol of democracy for alot of its recent history. As leader of the commonwealth the U.K also gets alot of help and funding from other countries. Its Universities are cheaper then Canada's and it also has a good health care system. Democracy ensures a higher quality of life then many other countries.
#3.Australia - A great country to live in (Excluding the recent floods) again alot of democratic values at its core. Australia also enjoys a decent climate and is a country I really want to visit.
Now most influential and most powerful can only be three countries.
#1. The United States , currently the most powerful country on earth due to its military might. The US also doesn't shy away from a fight forcing its views on any country that dare oppose it. Now I like some of these views like womens rights and democracy so I do like the states to a degree. Capitalism has also made American companies bigger and more wealthy then non american companies spreading their influence over the world. With military and cultural influence spread by capitalism , democracy and the US Army it is undeniable that the US is the most powerful country on earth.
#2. Russia , like the US Russia's military rules over its region crushing any voice that opposes it. Also Russia uses its abundancy of resources to influence Europe and several other nations. Russia also has communistic traits that remain despite officially becoming democratic which lead it to having influence in several countries which refuse to be influenced by America. Russia also has a seat on the Security Council and can throw its weight around to influence other countries. Also by allowing other countries to obtain nukes and nuclear technology they can prevent the US from controlling more of the world. Also by influencing countries like N.Korea and Iran it can avoid future conflicts with the US by keeping their military occupied.
#3. China, soon to be number one China is growing in power day by day. Right now China controls the worlds biggest manufacturing country. Meaning if China goes down so does the world. With over a billion people China has very big power over the global economy. Also the US owes China trillions of dollars and when China gets its money back it will be more powerful financially then the US or Russia. China is also advancing its military technologies and within decades could over power the US.
In the end I think my home country of Canada is the best. But since I couldn't choose Canada I gave my favorite countries as well as the list of the obvious super powers!