The basis for Aristotle's classification of constitutions is twofold, taking into acccount (1) that government may be exercised for the good of the governed or the good of the governing and (2) that government may be concentrated in one person, a few persons or the many (Politics, III.7).
The 'true' forms of constitution (orthai politeiai) are those in which the one (basileia - kingship), the few (aristokratia - aristocracy) or the many (politeia - 'polity' or constitutional government) govern with a view to the common interest - the good of the governed.
The corresponding perverted forms (parekbaseis) are tyranny (basileias), oligarchy (oligarchia) and democracy (demokratia). The hallmark of oligarchy is government of the rich, and of democracy the government of the poor, in their own interest.
Aristotle's order of preference for these six forms is: kingship, aristocracy, polity, democracy, oligarchy, and tyranny (Nicomachean Ethics, VIII.91160a).
He concedes, however, that the qualifications for kingship and aristocracy are seldom to be found and that, short of the ideal polis, polity is likely to be the best option in practice. Polity is constitutional government in which the middle class predominates. Ths class makes for stability since the members of this class 'do not, like the poor, covet other men's goods; nor do others covet thers, as the poor covet the goods of the rich; and as they neither plot against others, nor are themselves plotted against, they pass through life safely' (Aristotle, Politics, ed. S. Everson, Cambridge: CUP, 2010: 107-8).
How far this classification maps onto modern constitutional forms is open to question. To begin, are modern societies class-divisible into rich, median, and poor? Or do we need a more refined social categorisation? More than that, polity does not on Aristotle's account envisage representative government. The polis or city-state that Aristotle analysed and prescribed for was governed by its citizens directly and not through representatives. The citizens, moreover, comprised only a subset of adult males and excluded women, slaves, and foreign workers or traders whatever their length of residency or contribution to the economy.
A further point is that Aristotle had what Rawls would call a substantive conception of the good. This is outlined in the Nicomachean Ethics and, in brief, involves the pursuit of eudaimonia (or well-being, human flourishing) through the exercise of the ethical and intellectual virtues - specific traits of character and intellectual capacities which Aristotle enumerates and specifies. The common interest makes sense and has plausibility against this background; the common interest is to create and maintain the conditions for the achievement of eudaimonia. However, eudaimonia is fully attainable only by a minority of the population, namely the adult male citizens and probably in practice not all of those. Aristotle could rescue his notion of a common interest by arguing that in a polity those citizens who have attained eudaimonia have the interest of other citizens, and not solely their own interest, in view: and so the interest of all is served by citizens (of moral and intellectual virtue) best competent to rule. That 'all' included women, slaves and foreign workers and traders is, to say the least, far from guaranteed.
A final point is worth brief mention. A widespread current assumption is that democracy, under whatever precise form, is the proper political norm. Aristotle took a more flexible view. While in general and the abstract he clearly prefers polity as the best feasible option, he is sensitive to context. A central task of politics is to find the best constitution 'relative to circumstances' and 'given conditions' (ez huokeimenon, ez hupotheseos): Everson: IV.1, 1288b: 92. Circumstances and conditions might not lend themselves to the success of polity whatever its general and abstract merits. Specific circumstances and conditions trump the general and the abstractly preferable.