Webster 1913 gives a very general defintion of a confederation but fails to distinguish it, as a form of governance of several states under a central authority, from a federation. In a federation, each member state cedes a great deal of sovereignty to the central government, while in a confederation, the member states are very nearly independent members of a larger union existing for common identification, foreign policy and common defense. It is difficult to determine where exactly to draw the line, however. Elements normally existing a federation might be present in a confederation, and vice versa, if enough independence or ceding of authority is present in other areas.
There are quite a large number of Federal governments throughout the world, notably the United States, Germany, and Russia. Confederal governments, on the other hand, are few and far between. Switzerland is the only example of a country currently operating under a confederal system, but there also was a short-lived confederation of Senegambia between Senegal and the Gambia in the 1980s. (Thank you Gritchka.)
Two famous examples in history illustrating the conditions leading to the formation of confederations are the original goverment of the United States of America under the Articles of Confederation, and the Confederated States of America that existed during the American Civil War. Note the presence of "Confederation" in both names there, compared to today's usage of Federal when referring to the national government. In both situations, the nations in question were formed after thinking that their states were being controlled far too much by their centralized authority.
For the entire history of its American colonies, Great Britain ran (and still does run) as a unitary system, and Parliament has had the authority to make law at every level of government. There were legislatures in all the American colonies prior to independence, and they were in general loathe to reliquish power to any central authority which, like Parliament, might pass all sorts of laws and taxes without approval of those they affected most. Thus the Articles of Confederation purposely created a weak central government, and left most power in the hands of the states and their repsective legislatures. However, given that it couldn't even mandate taxes from the states, it was doomed to failure, and was replaced by a federal system.
Over the years, the nation slowly became split between the north and south. The economies and philosophies of the two sides were slowly drifting apart, slavery being only one issue among many. After the addition of more non-slave states into the Union gave the North a majority in both houses of Congress, the seceding southern states felt they no longer had a voice in the govenment, especially once Lincoln was elected President without a single electoral vote from the South. Since they were afraid of the US federal system that caused a tyranny of the majority, they set up their new government as a confederation in order for their main grievance to be rectified. It failed quickly because it was invaded (or liberated, depending who you ask) by its previous masters and forced (or compelled) to rejoin the federal union. We can only speculate as to whether the government would have been viable if the Confederation had cooperation instead of hostility from the rest of the Union.
At the heart of the confederal government system is the idea of states' rights. The colonies' and southern states' desires to do things their own way within their own borders without having to accept rulings from higher authorities made them dislike the unitary and federal systems and caused them to strive for a looser union.
As mentioned before, it is difficult to draw the line between federations and confederations because of differing circumstances. It is perhaps just as difficult to draw the line where a alliance of nations becomes a single nation under a confederation when looking at the developing European Union. It shall be interesting to see just how strong its government will get. Given that the EU has a common monetary system, all countries have ratified a treaty on Human Rights that supercedes their member states' laws (something which many British MPs do not relish), and the borders are extremely open, the EU is very close to being a confederation and thus in some way a single nation. However, requirements of a nation not present in the EU are a standardized foriegn policy, a common defense force, and most especially, the member states no longer considering themselves sovereign nations in their own right. Even if all the member nations agree to cede complete control of their military to the EU, and rely on it alone to determine foreign policy, it seems unlikely that the people of Europe will any time soon feel that their homeland no longer is a nation itself.