(Keep in mind that while I've studied economics informally for a couple of years, I'm only halfway through my first semester of formal study. However, I'm finding that my amateur intuitions are pretty accurate, at least so far.)
You, gentle reader, probably have some intuitions about economics, built from your ordinary life. You shop around and buy things, you have some sense about the wages you can expect for your labor, and you probably have a reasonable grasp of interest rates, both those you receive (for your savings and 401 (k)) and those you pay (for your loans and credit cards).
These intuitions apply mostly to microeconomics, the study of individual economic behavior within a national economy. In microeconomics no individual typically has the power to arbitrarily affect "global" or large-scale economic variables: your decision to buy or not to buy a television, for example, won't measurably affect the price of televisions.
One very difficult concept for people to grasp is that macroeconomics is almost totally different from microeconomics. Macroeconomics is primarily the study of the aggregate behavior of national economies as a whole. A national economy is an economy under a single more-or-less unified governmental control, usually with its own currency (obviously the EU and the Euro is an exception), which the government can arbitrarily control. (Note that just because a government can arbitrarily control a currency does not mean it can do whatever it likes without any consequences.)
One important difference is that microeconomics is all about relations between individuals (and individual companies), and how all those complicated relationships sort themselves out. A national economy, however, does not have any relations, there's nothing for it to relate to. (Not precisely true: how national economies relate to each other is yet another domain: international economics. But the vast majority of economic behavior for all nations is within the national economy, and the kinds of relations individuals have are very different from the relations that nations have. On the whole, it's easier to think of a national economy as being self-contained.)
One important illustration of the difference is one's conception of money. In microeconomics and individual finance, it's useful and valid to think of money as concrete and physical. I have this much money in the bank, I'm expecting that much money in my next paycheck, and when that money is gone, I can't buy stuff.
But money isn't really concrete. It's a socially constructed method of expressing relative demand. In macroeconomics, there's nothing for the national economy as a whole to be relative to. Hence the first thing I've learned in macroeconomics is to abstract money away, and talk about economic activity in "real" terms. The process of abstracting money away is complicated, uncertain, and provides only estimates. But, on the whole, and in the long run, actual dollars and cents, the actual numbers on the actual bills in your wallet and the numbers on your bank accounts are irrelevant to macroeconomics. I don't know anything if you tell me you have a billion dollars in your pocket: if you're an American, you're probably Bill Gates or Warren Buffet; if you're a Zimbabwean, you're going to come up short at Starbucks.
Which means that economic activities that are only about money, notably debt and finance, have an entirely different meaning in macroeconomics. For example, in macroeconomics, debt does not exist: every debt is one person's liability and another's asset, and on the whole they cancel each other out exactly. I don't mean that debt is meaningless or unimportant in macroeconomic terms, I mean that debt doesn't have the same appearance of concrete reality in macroeconomic that it does to an individual in microeconomic terms.)
The upshot is that most of your microeconomic intuitions about money, debt and finance are meaningless or incorrect when applied to macroeconomics.
Another way our ordinary intuition fails is in our treatment of government. In economic terms, government is really different. Governments act (or should act) not just differently but opposite from how individuals act. When times are good, the government should spend less; when times are tough, the government should spend more. And governments should spend on precisely those projects that are not individually profitable.
Government debt is also completely different from private debt. Governments do not have to forego consumption to pay back debt. Since governments do not labor or produce, a government cannot allocate some of its production to repayment of debt. Instead, government debt affects the economy as a whole. When an individual is "doing well" economically, it makes economic sense to borrow money (for suitable purposes), because she can expect the value of her labor to cover the cost of the debt; when she is doing poorly, it makes economic sense to save money and pay down debt. Governments, though, operate in the opposite manner: when on the whole times are tough the government should borrow money; when times are good the government should save.
We do this because when times are good overall, if one individual is doing well, they can "trade" with others who are doing poorly: the individual who is doing well borrows and buys from the individual who is doing poorly, enriching them both. When the economy as a whole is doing poorly, there are too few individuals who are doing well to absorb the savings from too many who are doing poorly. Everyone wants to save, but savings is relative: it is a transfer of demand from one person to another. If there's no one to borrow other people's savings, saving just depresses the economy as a whole. So the government must step in and borrow everyone's savings.
Another intuition that doesn't transfer well from microeconomics to macroeconomics is the idea of dynamic equilibrium. In microeconomics, pretty much everything progresses to the more-or-less optimal equilibrium "by default", without any need for global or high-level intentional action. Microeconomics is dynamically stable: small variations from the optimal equilibrium statistically "pressure" circumstances to return to the optimal equilibrium. Indeed the pressure is so strong that (with some exceptions) it is difficult to deliberately affect microeconomic behavior at a global level, and usually counterproductive to do so.
I'm pretty sure the dynamic stability of microeconomics is a part of standard economic theory. I strongly suspect (but do not actually know) that macroeconomics is dynamically unstable: a small change to the economy as a whole statistically "pressures" the economy as a whole not to return to an optimal equilibrium but rather continue to move in the same direction as the small change. A growing economy tends to cause the economy to grow faster; a slowing economy tends to cause the economy slow more. Inflation causes more inflation; unemployment causes more unemployment. Managing a macroeconomy is like balancing a pencil on its point.
Overall, I think it's useful to keep in mind a model of the village: a small, self-contained community of individuals who exclusively trade with each other. Microeconomics will tell you about the relative activity of individuals within the village, macroeconomics will tell you about the activity of the village as a whole. What does it mean, for example, for the village as a whole to borrow from its own members? Clearly it means something very different from one individual borrowing from another individual. Test any economic concept in your mind against this model: what would it mean for an individual to do something? What would it mean for the village as a whole to do something? If the concept makes sense in one context but not another, you've hit on a micro vs. macro difference.