The word "regime" gets tossed around a lot nowadays without anybody really thinking about what it means. It means something quite different to a particular state, nation, or government. If you think about it, the etymology is clear - you can have a diet regime, a health regime, and a political regime. It's a way of doing things, a "regimen", ultimately derived from the Latin regimin.
A political regime is hence a set of rules or ways which politics are done in a particular country at a particular time. A state is quite different from a regime. There has been a French state for a long time, but there have been numerous different regimes which have acted through and modified this state. When a drastic change of regime takes place from within, it's properly referred to as a coup d'état, even though we usually nowadays only use this phrase for the installation of a new regime by the military (see my write-up for more details).
Of course, nowadays the word "regime" is usually used to refer to a regime which the speakers disapproves of, like "the Iraqi regime" before the war or "the Iranian regime" at the moment. I don't think I'd be too far off to claim that this is mostly to do with the fact that political philosophers who have been popular with our right-wing politicians and intellectuals are the ones who continue to use this term (in the original, non-judgemental sense) - Leo Strauss and Allan Bloom spring to mind. This in turn is beause these are the only political philosophers who continue to seriously engage with the classics, from where regime classification as a concept initially springs.
Meanwhile, on most of the left and in the center, the word has fallen out of use because it tends to highlight what is different in political systems rather than their similarities. Such an effect is clearly inimical to, for instance, the project of the United Nations, a democratic body which regards Sweden and North Korea as equal "member states", despite their incredibly different regimes. Many Westerners carried from imperialism the primary lesson that the West ought to stay out of the rest of the world, and so the ideas of unrestricted sovereignty and non-intervention were invented in the West. The idea of a regime, which presupposes a judgement on exactly what that regime consists of is dangerous to this global system - and so it is not a word that finds much favour amongst the left.
Anyway, regardless of who employs it, it essentially refers to both the form and content of a political system. By the form I mean the particular way power is distributed amongst various branches of government and organizations. Some of these forms are incredibly routinized and have stayed essentially the same for a very long time - for instance, the American regime can be said to have remained essentially the same since its founding. But in the Nazi regime, the centers of power were constantly moving around, at one time elevating this group, at the other denigrating that. The organs of power are probably the least interesting part of a regime, but they do determine how it functions on a day to day basis.
Much more interesting is the content, which really determines the form anyway. The content is about who rules and on what principle. To the Ancient Greeks, there were three main forms of pure regime - oligarchy (rule by aristocracy), democracy (rule by all), and monarchy (rule by a king). You can see here the relationship to the form - the United Kingdom might abolish the House of Lords, and it would still be a democracy, but this action would mean the start of a new regime. Anyway, our political experience has expanded somewhat since the time of the ancients, as has our understanding of exactly what "politics" is; hence, to these three types and their mixture, we might add the totalitarian regime and the dictatorship (the Greeks knew about tyranny, but their understanding of the term "politics" did not encompass it).
But even this doesn't tell is all there is to know about a regime. We need to know from where it derives its authority, as this largely determines its character and the sort of person it tends to hold in the highest esteem. You can judge a regime by the human type which it regards as the highest. So, in the city of Socrates' speech, the philosopher is the highest human type; in an aristocracy, the military hero; and in a totalitarian dictatorship, the individual who is most unthinkingly subordinate to the Party is highest. This is because these regimes derive their legitimacy, repsectively, from philosophy's victory over politics, from the rule of the virtuous and best, and from the complete rightness of everything that the Party does. A dictatorship cannot rule a nation consisting entirely of rational, economic utility-maximizers, but would have to transform them or itself.
This leads me, if only briefly, to the idea of "regime change". You can see where this idea comes from - the change of the set of rules which governs politics in a country from, say, a dictatorship to a free-market democracy is indeed a regime change. But it tends to ignore the close identification between regime and society, or the people who make up that regime and are subject to it. The regime is not just the couple of dozen people sat at its apex, but an entire system of authority and categories of legitimacy and thought which exist in people's heads and hearts. Even if one accepts the moral superiority of democracy to dictatorship (which one must), this in no way reduces the practical difficulties in bringing about such a change; and "regime change" means changing all this, not just decapitation.