The Union is said to be subsidiary to member states and can act only in those areas where "the objectives of the intended action cannot be sufficiently achieved by the member states but can rather... be better achieved at Union level." The principle is established that the Union derives its powers from the member states.On the one hand, this would seem to be a stronger limitation on central power than we have in the U.S. First, it doesn't just give the EU power to act, but explicitly limits the circumstances in which those powers can be used. It would be as if the U.S. Constitution said Congress could regulate interstate commerce, but only when the states weren't capable of handling the problem themselves. This, I think, would cut against (among other things) the rampant federalization of criminal law in the U.S., where ordinary drug dealers, gun owners, and gang members become federal felons. Also, the EU is an organization created by true sovereigns and derives its powers from the agreements entetered into by the member states; our constitution certainly treats the states as important and quasi-sovereign entities (e.g., amendments must be ratified by 3/4 of the states, each state gets the same number of senators irrespective of population, etc.), but the federal government's power is at least in theory derived from the people.
On the other hand, Europe doesn't have our tradition of robust judicial review--in the Netherland, for example, the Supreme Court cannot strike down a law that conflicts with the Dutch Constitution--and it's not clear that the subsidiarity principle will be an effective check on the tendency of EU officials to regulate more and more areas of life and commerce. The elected European Parliament is toothless (though the constitution would give it some additional power), and the European Court of Justice has much more limited powers than our federal judiciary, leaving the unelected executive (the EU Commission) pretty much unfettered.
Unlike the American states in 1787-89, the Europeans are not holding ratifying conventions; their governments will accept the constitution or not in the usual way they ratify international treaties. But several member states are holding referenda. The constitution scraped by in its first test in Spain, but it may receive a fatal double-blow in two weeks.
There is no nation more fanatically pro-Europe than France, but it looks quite possible that a majority will vote against the constitution in the French referendum on May 29. Three days later, voters here in the Netherlands may reject it as well. Clear majorities of elected officials in both countries support the constitution, but the voters will have the final say. The Dutch referendum is non-binding, but the parties in the governing coalition here have vowed to respect the result if there is at least 30% turnout. The French referendum is binding. The constitution requires the unanimous consent of the member states, so one failure will doom it for the time being.
If it's rebuffed this time, the constitution will be back in some revised form before too long. But the politics of the matter will have been changed significantly, especially if it's the French who turn it down. The French referendum is thus a critical turning point for the future of the EU, and therefore for the future of the world--including the U.S. Since I assume the American press has been inundating y'all with stories about the referenda, I'll desist from boring you with further details.
Update: corrected some factual errors in the original, most notably that I previously reported incorrectly that the French referendum was non-binding.