The Congress of Vienna is the part of the peace settlement at the end of the Napoleonic Wars that everyone always remembers the most, although there were also two Treaties signed at Paris and something called the "Holy Alliance" was formed on the suggestion of Tsar Alexander I.
The exact territorial juggling is not particularly interesting unless you're interested in the century between this and the next European general war, World War I. However, several things are of note about the Congress. First is the very fact it was nearly a century until there was another general European war, despite the slow disintegration of the Ottoman Empire and developments such as the rise of Germany. Of course, the settlement cannot take all the credit for this, and it could equally be argued that the settlement laid the way for the rise of Germany and Europe's thirty year catastrophe. However, this was before we learnt our lesson after World War II and moved into a post-modern state structure.1 Practically any system trying to operate on the "balance of power" was bound to collapse eventually, and the wonder is it took so long.
The architects of the peace settlement are often censured for not taking liberalism and nationalism into account when forming the settlement. As a particularly astute commentator has pointed out, this would have been like the post-1945 settlement taking Nazism as its guiding principle. Liberalism and nationalism were blamed for the twenty years of war that had just occured - like Nazism, they had burst forth from their homeland (in this case France), swept across Europe destroying all other states in their path, and worse yet they had found adherents in nearly every country they conquered. To now enshrine these dangerous and subversive ideas as permanent fixtures of Europe would have struck any right-thinking man as insane.
Indeed, the Congress was mainly concerned with leaving things just as they had been in 1789. After Napoleon had been banished to Elba in 1814 and before he returned for his last stand, the victors did everything to conclude an honourable peace that would be long-lasting. Unlike the Allies at Versailles in 1919, they first ensured they had someone they considered credible to talk to - by putting Louis XVIII on the throne in Paris. In the first Treaty of Paris, France did not have to pay an indemnity, she made territorial gains, she got to keep the art she had looted from all over Europe and she was invited to Vienna to help redraw the European map. Then Napoleon returned and the Hundred Days began, ending in the Battle of Waterloo.
The second Treaty of Paris was much harsher - a 700 million franc indemnity, troops stationed around her borders (which she had to pay for), and a return to her 1789 borders, plus the return of Europe's art. The apparent instability of France seemed to suggest a tighter leash was necessary, although she was soon admitted back into the comity of nations.
With France apparently subdued, much of the Congress considered of common power politics between the Great Powers. Although every sovereign of Europe was represented, in reality Austria, Prussia, Russia and Great Britain had decided beforehand in a secret protocol that they would make all the decisions. Most of it took place in informal meetings between dignitaries from these countries, especially Viscount Castlereagh of Britain and Prince Metternich of Austria. Both men have been much maligned, both on their own terms and in the case of Castlereagh for mere association with the latter, but in the latter half of the last century their reputations were much restored, not least due to a book by Dr. Kissinger which grew out of his doctoral thesis.2
Metternich and Alexander I may be considered more justly maligned from the standpoint of a Western liberal. Of the various alliances formed as part of the peace settlement, their favourite was the Holy Alliance. The British preferred the Quadruple Alliance, an association of states that existed merely to make sure no-one messed with the Vienna settlement and tried to redress the "balance" (often understood as a territorial balance, i.e. the territorial settlement). However, the Tsar had different plans - he wanted to use another group, the Holy Alliance, to maintain the domestic status quo in Europe as well. This meant that if there was a revolution in, say, Spain, the rest of Europe would send the troops in to restore the rightful ruler and then depart. Almost everyone in Europe signed up, except for the British who pleaded constitutional difficulties.
In reality, Castlereagh didn't want to sign - he thought it "a piece of sublime mysticism and nonsense" and unenforcable if even desirable. The British were proud of their own tradition of non-intervention in the affairs of others over the next century, but in reality this was often a principle born of necessity due to Britain's constitutional difficulties with keeping a standing army. Castlereagh preferred to work through the Congress system to discuss matters with other powers to find peaceful solutions to problems, something that was largely successful for some time to come. The peace settlement was not perfect and was in time subverted, but its weaknesses would pass on a legacy of lessons to its improved successors.
1The term is Robert Cooper's in The Breaking of Nations: Order and Chaos in the Twenty-First Century (London, 2004). He argues that the world is split into three parts, a pre-modern, a modern, and a post-modern. The post-modern part, basically integrated Western Europe, has banished war by redefining sovereignty as a "seat at the bargaining table" of an international organization and by sharing strategic resources.
2Yes, that Dr. Kissinger - it's A World Restored: Metternich, Castlereagh and the Problems of Peace, 1812-22 (Weidenfeld & Nicholson, 2000)