The first way to characterize the form of a state is by means of the form of sovereignty. What is meant by this a way of describing the different persons that have supreme power in a state. Either there is only one person who has supreme power (autocracy), a group of people possess it (aristocracy) or all do so (democracy).
The form of sovereignty is distinct, on Kant's account, from the form of government. The form of government concerns the constitution of the state in question, that is, the way in which the government operates. Whilst Kant distinguished three forms of sovereignty he suggests, by contrast, that there are only two forms of government. The two forms of government are republican or despotic. "Republicanism is the political principle of separation of the executive power (the government) from the legislative power; despotism is that of the high-handed management of the state by laws the regent has himself given, inasmuch as he handles the public will as his private will." (Ak. 8: 352)
In characterizing republicanism in the way suggested in the above citation Kant is following the examples of Montesquieu and Locke. The separation of powers involves the distinction, at a minimum between executive and legislature (though it would be usual to make clear that each of these is distinct also from the judicial power). In endorsing this conception Kant indicates that this separation is precisely what enables there to be a form of public will. Without this distinction the ruler will usurp the possibility of public will and make everything his private domain (as suggested in the earlier posting about corruption thus indicating that a despotic state is inherently corrupt).
After distinguishing sovereignty from government Kant proceeds to make clear the basis for the difference between republicanism and democracy since, not only is democracy a form of sovereignty and not a form of government (unlike republicanism) but it is also the form of sovereignty most allied with despotism. The rationale for this claim that Kant gives however makes clear that his conception of democracy equates it with majoritarian rule since he describes it as one in which "all decide for and, if need be, against one (who thus does not agree)", a conception that provokes the view of a general will in contradiction with itself. In viewing democracy in this essentially majoritarian way Kant's conception of it is very similar to that of Plato.
Since contemporary "democracies" do not operate by the simple conception of all deciding for all that Kant refers to but rather through mechanisms of representative government then they are close in conception to his view of republicanism.
The conclusion of the discussion of the first definitive article repeats the view that the distinction, when it touches on forms of government, is between republicanism and despotism and asserts firstly, that republican government is intrinsically representative in nature and that despotism is, by distinction, a formless mode of government. Having indicated that democracy, on Kant's conception of it, is intrinsically despotic, the point is subsequently made that autocracy and aristocracy, are at least possibly able to be representative. This includes a reference to Frederick II's claim to be the highest servant of the state.
So attached is Kant to the sense of representation being what gives a government form that he further suggests that since the smaller the number of people involved in governing the state the more representative in nature it is so the autocratic form of sovereignty is best suited to the republican end. This claim however could as easily be used to defend either a prime ministerial system (where the executive is concentrated in one person) or a presidential one. Under Kant's account both would count as forms of republican autocracy.
The kind of sovereignty is, however, much less significant than the form of government and it is the latter that is either republican or formless (hence corrupt and intrinsically at odds with the very notion of the public). The ancient republics are finally condemned as not real republics at all because they did not know of the separation of powers (and thus were despotic in principle).