A public good is something that is of benefit to multiple people and that you can't prevent them getting the benefit from. Most things in economics are scarce - only I am able to enjoy the delicious citrus taste of the can of Diet Coke with Citrus Zest that I'm drinking now because my consumption of it prevents you from drinking it as well (get your own). A public good is something that everyone gets the benefit from, like national defence. This creates the problem of free riders, who get the benefit of public goods without paying for them - say, a tourist who goes to the seaside and throws his can of Diet Coke with Citrus Zest onto the ground, to be picked up by a council worker who his taxes don't fund.
Where the concept gets a bit more controversial is in international affairs, where we have the concept of a "global public good". Global public goods are things done by particular countries that benefit everyone else even if they don't have anything to do with it. Because they're inevitably provided by the dominant power and people are rightly suspicious of claims of altruism in foreign policy, I'm sure some people would say the concept doesn't exist. But it totally does.
An often-cited example of a global public good was the way that the United Kingdom kept the seaways open during the period of the Empire, allowing trade to continue between all countries unhindered. Of course, we Brits didn't do this just because we're jolly good sports. We shouldn't dismiss the belief in one's altruism as a motivating factor in foreign policy; but of course such beliefs are powerless against material interests which dictate to the contrary. Keeping the seaways open benefitted Britain, which was of course the primary reason we did it: but it's also indeniable that it helped the rest of the world, too.
A more recent example is the way that the United States has guaranteed the flow of oil from the Middle East. This priority gets a lot of stick from the "No blood for oil" crowd, but I'm afraid it's a simple fact that our economies need oil and have done for a long time. The main fear in the Middle East has always been that one power would come to dominate the whole region and be able to manipulate the price - and during the Cold War, people were worried that the Soviet Union had their eyes on this prize. The United States has stopped this from happening and so maintained the flow of oil; this is a global public good.
In 1979 Jimmy Carter declared the Carter Doctrine, which says that any "attempt by any outside force to gain control of the Persian Gulf region will be regarded as an assault on the vital interests of the United States of America, and such an assault will be repelled by any means necessary, including military force". That's why the Soviet invasion of Afghanistan was so scary - it looked like they were aiming at the oil fields. Then, when the Cold War was over Saddam invaded Kuwait. What was especially stupendous about the decision from Saddam's point of view was that, during the Iran-Iraq war, he had been a tool of the Carter Doctrine but apparently entirely failed to internalize it. Hence, he set about what appeared to be a first step towards controlling all of the region's oil (Saudi Arabia was next on his list), and the U.S. co-ordinated and directed the response and drove him back from whence he came.
No-one else, of course, was in a position to do this - and so the U.S. was providing a global public good by ensuring the supply of oil for everyone through their own actions. Of course, they did it because it benefitted them; but it also helped everyone else. And that's a global public good. We ought to judge superpowers by their capacity to provide global public goods, and it is certainly a criteria by which they judge themselves; the demand of the Russians to have a say in the Middle East peace process stems from the prestige they get from working on these issues of common concern to the world - on global public goods.
You and I are living in the era in which we will see the end of unipolarity and a new era of bi- or tri-polarity as Russia reasserts itself and China grows in power. A world order constructed by the Chinese and Americans together is going to be quite different to one just constructed by the Americans, as our current one is, because it was the US who created all these multilateral institutions - global public goods - that the Americans are chided for not working through, such as the United Nations, the International Monetary Fund and the World Trade Organization.
Despite its faults and excesses, America has underpinned the global system of trade, sovereignty and security for decades now - these things didn't just happen by accident. It has provided many benefits to free riders the world over. History will judge if the United States, overall, did more bad than good. But there's no God-given assurance the alternative will be any prettier.