The idea of a true United States of Europe goes back a good ways. Where you mark the beginning depends in part on where you put the dividing line between attempts to conquer all of Europe and attempts to unite it. Napoleon envisioned a united Europe, but he envisioned it as an empire which he rules, though with varying degrees of local autonomy. During World War One, many writers imagined that once the war had purged Europe of militarism, the way would be ready for a united Europe. French soldier Eugene Emmanuel Lemercier wrote in a letter to his mother in 1914:
Nov. 15, 1914: “Let me go a little into details concerning my views of a better future to be brought about by this war. These events are preparing the budding of a new life -- the United States of Europe. When this conflict is ended, those who have performed all their filial duties towards their country will find themselves brought face to face with still greater responsibilities, which cannot be realized at the present moment. But here is the paramount duty -- to try now to make the future secure. They must stretch every muscle to do away with all the causes of trouble between nations. The French Revolution, notwithstanding its shortcomings, certain backward steps in practical things, some weakness in its constructive measures, nevertheless impressed on humanity what was meant by national unity. Now the horrors of this war must do the same thing for European unity, for race unity. This new condition can be brought about only through suffering, spoliation, contests for years to come; but there can be no question that the door has been opened on a new horizon.”(His posthumously published letters can be read here.)
After the war, however, it proved that far from burning away the last obstacles to a United Europe, the slaughter had created new resentments and barriers to such a union. After World War Two, which was in many ways simply a continuation of the first, there was a desire for a united Europe among the liberal leadership that took charge of the Western democracies. However, this time things moved very slowly. Things took several steps around 2000 when the Euro was introduced, passports were no longer required for travel between EU countries, etc. However, although the EU has a single currency and a united banking system, it definitely is not a union on the order of the United States of America, and this seems to be creating tensions that could well destroy it. Understanding this requires thinking about the various components that go into a modern union such as our own, as it's developed during the 227 years since our constitution was written and the 150 years since our Civil War was won.
We have a national currency, the US Dollar, which our government backs by its faith and credit. That currency is used by the US to collect taxes and to pay for anything the government buys (wages, goods, etc.) The states all use that currency as well.
Since the federal government issues the currency, it has the ability to inflate the currency by issuing more than it brings in. However, it usually does not do this. Instead, it borrows money. Since the US has a very good history of paying on its debt, we're able to borrow very large amounts of money at very low interest. The states do not have the option of printing money and inflating the currency, and they are also prohibited from declaring bankruptcy by the contract clause of the Constitution and by the US bankruptcy code. As a result, states potentially have to take draconian cuts to staff and services if their income falls significantly short of our their outlays. The federal government, however, can mitigate this to a great extent in the modern US, because total Federal spending is greater than state and local spending combined. Many government programs and services are paid for directly or indirectly by the Federal Government. This means that while states may have to take severe cuts if they run into financial problems, there's a limit to just how bad things can get for citizens within the state.
This works in part because there's a fairly strong feeling that "we're all Americans" and so while there are arguments about how much the government should be spending, and occasionally people will put together maps of "giver" and "taker" states based on the ratio of federal taxes collected to federal spending by state, people don't actually get incredibly upset about helping our fellow Americans. Currently the state with the highest unemployment is West Virginia at 7.2%, while the lowest state rate is Nebraska at 2.6% (source), but the taxes of people working in Nebraska help cover the unemployment benefits of those not working in West Virginia.
The EU is a much less tight union than the USA, and people seem to think of themselves as French, German or Greek first and as European second. The structure of the union reflects that. There's one currency, and you can travel from one country to another and work in other countries within the union without needing a visa or passport. However, spending is pretty much all done at the state level, not at the EU level. Thus, although the German unemployment rate is 4.7% and the Greek unemployment rate is 25%, Greece needs to figure out how to solve its own unemployment problems. German taxes do not go to pay Greek unemployment benefits.
In addition to using a common currency, the governments of the member states all get to borrow money on common terms, but in return they're expected to behave with common responsibility. During the first decade or so after the currency was put in place, it seemed like this was going pretty well, and the Greek government took advantage of its new borrowing terms to fill the gap resulting from a social democratic sized social safety net combined with a culture of tax evasion. (They even covered up how bad their debt ratio was getting, allowing them to get further into debt than they should have according to EU agreements.) However, when the global economy hit the skids, weak economies such as Greece was things slow down much more than countries like Germany, and so Greece saw its already large gap between tax collections and government spending balloon as more people needed government benefits and fewer people paid taxes. (After all, when 25%+ of your workforce is unemployed, there are a lot less people to tax.)
With their own currency, Greece could have devalued their currency, making their exports more competitive in other countries and making tourism in Greece more attractive to foreigners (both of these because Greek money would be worth less compared to the currency of other countries.) However, with the Euro that's impossible for Greece and they're effectively stuck. They can't use inflation to kickstart their economy and their tax and spend system seems to have gone into a death spiral where people don't pay taxes because they're making no money, and because everyone is out of work they need government money to help people out.
To make they conversion, they would need to freeze everyone's financial assets and forcibly convert them into Greek Drachmas (or whatever they decided to name the new national currency.) Of course, people wouldn't want to have to make the switch, because the value of the Euro will doubtless remain fairly stable, while the whole point of switching to the Drachma would be to allow its value to fall at first in order to reset the Greek economy. So if the vote on austerity is "no", it's possible the Greek banks will remain closed until the new currency is declared in order to prevent a run on the banks.
Of course, it may also be that the Greeks will vote to accept the austerity program that will allow their government to stay in the Euro and keep borrowing.
Either way, this serves to underscore the difficulty of easing into a union such as that of the United States. The nations of Europe have a lot of differences and independent history which naturally make them reluctant to enter a union as close as that of the USA. However, by trying to have a currency union similar to that of the USA while not having the centralized taxing and spending authority of the Federal Government, they've created a whole other set of problems which may cause the union to splinter or at least shrink down to a couple core countries with similar economic strengths.