Since everyone else is doing it...
In all honesty, I can't limit it to ten. Our "world" is so multi-faceted that I can't really rank these books, so I won't bother numbering them.
The Republic of Plato. There is no one in the Western World who has not, in some form or another, been influenced by the works of Plato, and a great deal of political philosophy can find a prototype expressed in the Republic. As with many of the truly Great Books, however, this one is extremely dangerous. Plato searched from the one true Good in the universe and according to the story told in this book a perfect city could be made if only the very wise possessed unfettered power. I highly doubt "good" is unitary and I know that no human is wise enough to be telling another how to live his or her life backed up by the threat of the state. Plato himself seems to have known that; if only modern day statists could figure it out.
Federalist by Publius: This work has had a rather amazing story behind it. For those who read it while thinking about the situation in which it was written, one has to wonder if the authors would have been surprised how we almost revere this collection of essays today, long after the issue of creating a federal government has been settled. The true value of this book lies in the philosophy of man and his imperfections; they never sought out to cure those imperfections but only to limit their abuses in government itself, namely by dividing power so much that no one person or faction could accomplish their goals without compromise. A true warning for modern times.
Anarchy, State, and Utopia by Robert Nozick. A response to John Rawls argument that the state is needed to fix perceived injustices in the distribution of goods and talents by nature. Nozick does an excellent job showing the advantages of a minimal state, the necessity of the rule of law being applied equally to all, and the impossiblity of "social justice" (which always means different things to different people). A must read for anyone who actually desires to study the proper role of government rather than just its forms.
April, 1865 by Jay Winik. This should be read by more high school students. The Civil War did not just "end" in April with everyone saying "yup, it's over, might as well go grab a beer and head home." Even with the Confederacy on the verge of collapse, it was by no means certain the fighting and the killing would end with the destruction of the political state. Guerilla war, similar to what we saw in Iraq, was rampant in places like Missouri, and some of Lee's officers recommended continuing the fight by such means rather than surrender. Lee, Lincoln, and Grant were all amazing commanders in their own way, but each showed their highest quality at the end.
Crime and Punishment by Fyodor Dostoevsky. I love Russian literature and Dostoevsky is a favorite. The conscience of a human being can be a dreadful thing, something I know from first hand experience but preferred immensely more from reading this book.
The Proper Study of Mankind by Isaiah Berlin. My introduction into the world of political pluralism. Berlin's understanding of the imperfection of man, even (or perhaps, especially) those in power, is a lesson I wish I could magically implant in the mind of every person. He also dismantles the idea that "the good" is unitary.
Second Treatise of Government by John Locke. The intellectual founder of our Republic. A limited government based on the consent of the governed with the purpose of protecting our rights (in particular property rights)...how radical he would seem nowadays!
The Great Divorce by C. S. Lewis. Though I am not a Christian any longer, C. S. Lewis will always be one of my favorite authors, and his explanation of personal choice in this book was nothing short of amazing. Mankind truly is petty.
Atlas Shrugged by Ayn Rand. I picked this book up in January of 2009, when America was taking a distinct turn towards statism. Rand predicted the failure of governments running economies before people like Barack Obama were even born, but apparently nobody every lent poor Obama a copy. More important than the technical aspects of the impossibilty of socialism, though, is the moral superiority of the free market; socialism treats men as cogs, capitalism treats men as men who are capable of determining their own lives for good or ill.
The Essential Writings of Ralph Waldo Emerson. The Prophet of Nature and Man. His beautiful essays cover a vast realm of topics but always manage to focus on the pure and beautiful. While there are aspects I disagree with, in particular concerning the unitary nature of everything, I still always pick up this book when spring time comes around to help enhance the wonder of the season.
The Bible. Nobody knows exactly how long it took to write the Bible or how many people were involved, but it does seem safe to say it spans hundreds of years with dozens of authors. No book has influenced Western literature as much as this one and I highly doubt there is another which influences personal morality as much, either. The stories and themes are wonderful and far too many to list.
1984 by George Orwell. I end this list with the two most abused books in history. Orwell was a magnificent author, but more importantly he was an honest one. He looked straight into the heart of mankind and saw its desire for power, for money, for lust, the weaknesses people carry with them, but the true horror comes in the form of domination of life, and nothing is as capable of dominating life as an all powerful state. I seriously considered adding Reflections on a Ravaged Century by Robert Conquest here, which focused on the horrors of the 20th Centuries totalitarian states, but Orwell's fiction will always be more memorable than reality.
Anyway, that's my (incomplete) list of books that have significantly shaped my world view, whether on God, politics, society, ethics, economics, or any other such field.