A useful approach, I think, is to look at the notable categories.
- Western style legal and political systems in the tradition of either England or Continental Europe (e.g. South Korea, Argentina, India), with (e.g. Belgium) or without (e.g. France) a constitutional monarch.
This said, it isn't entirely categorical. The first draft of most of the East Asian countries with Western style legal and political systems often have literally translated wholesale more or less paragraph by paragraph the formal legal codes and procedural rules of European legal and political systems. But, despite having the same exact same blueprints on paper, the way the system works in practice in a place like South Korea or Japan is profoundly different, mostly because the way human beings interact with each other in these transplanted systems is deeply infused with Confucian ideology that is so second nature that it isn't even remarked upon.
Another notable cultural overlay that profoundly influences how a Western style system plays out in practice is India where caste and ethnic distinctions and deep historical practice and custom pervade a superficially British style federal political system. Contrary to modern revisionists who attribute caste practices to British racist patriarchy influences, this political and social organization was even more distinct from the time that genetic evidence indicates that jati boundaries hardened ca. 1000 CE until British influence became pervasive ca. 1600 CE. The pre-colonial caste system of India is probably the closest a society has come in real life to the Divergent books, although the guild based democratic city-states of Europe also have some similarities to it. A less elaborate and simpler caste-like division of labor was present in the Indo-Europeans who expanded across Europe, West Asia, South Asia and Central Asia (especially in the Bronze Age).
Traditional Confucianism While there are really no states organized this way any more, historically, China and other states in East Asia were organized on a Confucian model. This superficially looks like feudalism but with more reciprocity between superior and inferior and historically a meritocratic system of leadership selection in contrast to the pure nepotism of clan based systems and hereditary systems discussed below. Confucianism emerged from a long standing political debate in China between rule of man (seen as superior since not prone to being twisted and focusing on selecting morally superior leaders rather than regulating them with rules) and the rule of law (seen as inferior due to the potential for manipulation of law by immoral tricksters and harsh morally). This dimension of rule of man v. rule of law is a useful dimension upon which to evaluate and categorize political systems more broadly. Some of this is an overlay on cultural differences between Northern China whose historically millet farming economy developed more individualism than Southern China's more communitarian historically rice farming economy.
Absolute monarchies (e.g. Saudi Arabia) or constitutional monarchies with significant real royal power (e.g. Morocco, 18th century England). A more complex version is feudalism in which there were landlord-tenant-like relationships between layers of aristocrats (in England, as a legacy of land grants in the wake of the conquering Norman army ca. 1066 CE and carrying over what was initially a martial law system). Feudalism wasn't just for Europeans. It was basically the structure of Japan and China for much of their history. Another interesting aspect of this is the balkanized patchwork system of non-contiguous principalities that were in personal union with each other by having the same hereditary leader in early modern Europe through the early 19th century. Malaysia has a rotating kingship among several regional kings. Bhutan was formally a monarchy for much of its history but is unique is many, many ways in how it organizes itself politically and culturally.
Military dictatorships There are lots of these. Most are short lived. Some are basically Western style regimes minus election based regime change. Others are totalitarian. Even Western countries have military based or occupied territories during wartime that are run this way (e.g. the occupied South during Reconstruction in the U.S.).
One party states on the Soviet Communist model (e.g. Belarus or the former Soviet Union). One key feature that is particularly with noting is that dual and parallel political structure. Formally, these states have similarities to organizations on a Western model. But there is a parallel party structure which so pervasively influences what goes on within the formal system that the formal structure is almost meaningless when it comes to policy making. This is also true of Chinese Communism.
One party states on the Chinese Communist model. This one is really important, because Soviet style communism is an extreme but more or less direct offshoot of European civil law political thinking. The Chinese Communist model looks similar at the forest level, but once you get down in the weeds, you see that a lot of the core, taken for granted assumptions of European politics and law are absent. Chinese Communism as it has developed is for want of a better description less black and white. For example, suppose that a man yells at his neighbors and acts like a jerk but pays his bills, shows up to work every day, and doesn't steal or kill. In the Western mindset there is a binary decision to be made, we can intervene or we can't intervene as government. In the Chinese Communist system the distinction between the personal and the governmental is mushier. The neighborhood party boss can invoke mild but governmental involvement in this guy's situation without him having to cross any particular line. Similarly, property rights aren't all or nothing in Chinese Communism. In Chinese Communism your home isn't your castle, it is more like your well established campsite that doesn't get the same level of respect but also isn't something that is meaningless, but isn't something has to be abolished completely either. And, while the org chart of the Chinese Communist government looks a lot like any Western bureaucracy, if you magnify more closely you find that there are a lot more generalists and there is a lot less specialization in how those agencies are run. Chinese Communism is also notable in practice for the fact that its wealthiest individuals overlap almost perfectly with its top political leaders.
Theocracies (e.g. Iran, the Vatican). Iran is another really interesting case because it too, while more within the Western political and legal model than Chinese style Communism, is still sufficiently innovative that it really needs a box of its own. The political organization of the Roman Catholic church embodied in a small sovereign state at the Vatican is another outlier that is highly unique (for example, with continuous non-hereditary succession through the College of Cardinals and its own auxiliary language, i.e. Latin). A third fairly unique but much smaller scale system is the way that the polygamous Mormon communities (e.g. one near the Arizona-Utah border) is organized. A fourth is the organization of the Jewish communities in the diaspora, such as New York City's ultra-orthodox communities. A fifth is the organization of the American Shaker communities. A sixth is the pre-Chinese conquest Tibetan Buddhist theocratic regime.
Shared secular and religious leadership The example of a religious establishment and a secular ruler who share power is common. See, e.g. Saudi Arabia, Sumerian city-states, ancient Egypt, feudalism co-existing with Holy Orders in the Middle Ages (compare and contrast the Benedictine order, the Franciscan Order and the Jesuit order with different modes of interaction and strategies), established Buddhism in East Asia and Southeast Asia, classical Greco-Roman and Semitic pagan societies, and the established Catholic or Protestant churches of early modern Europe. Neither has absolute power. Bali in the Middle Ages also comes to mind (with a Hindu religious establishment), as does pre-modern Indian for a significant part of its history.
Islamic systems (e.g. Afghanistan and Iraq and ISIL). Another overlay like that of Confucianism on a Western style model that profoundly impacts how systems play out in practice is that of systems that aspire (and succeed to a greater or lesser degree) to recreate the merged church and state Caliphates of the heyday of the Islamic Empire. The Shi'ite system is more familiar in its hierarchy. The Sunni legal system is radically decentralized (in tradition bearing organizational similarities to the organization of Rabbinic Judaism as an institution). Sufism likewise is radically decentralized.
The French revolutionary regimes. Several of the regimes that replaced each other in quick succession in revolutionary France in the 18th century, such as the Paris Commune, were particular notable, but obviously, not very stable.
The Medieval Iceland system. In this system there were a dozen or so chiefdoms in Iceland from which people could move freely from one to the other and a central government with a legislative branch and bodies convening to determine who is at fault in disputes, but no executive branch. If the Allthing determined that someone was a murderer, the remedy was to righteously gather up your loyal supporters and hunt the guy down yourself with official approval.
Clan systems (e.g. Somalia, early modern borderlands Scotland, pre-monarchy Saudi Arabia, Roma clans in Europe). In a clan system your political system is wrapped up into your extended family, you marry your cousins, and at the top of each clan is a warlord. There is no overall authority over the entire clan based political community and the state is weak and heavily influenced by nepotism. A lot of modern criticism of anarchism argues that in the absence of a strong state that the natural state of nature looks more like a clan system and less like a libertarian paradise. This was the way that the Huns and the Mongol Empire were organized and probably played an important part in the groups of expanding Indo-Europeans like the Mycenean Greeks, the Bell Beaker people, the Corder Ware people, the Avestians of Iran, and the Indo-Aryans of India. It was also how the early Hebrews were organized. The Roman urban tribes are another example of this particularly relevant to Divergent (and a bit like the Harry Potter "House system" on steroids. Most modern day Iraqis have a tribal affiliation that is arguably greater than their affiliation to the Soviet style communist state that existed prior to the Iraq War (particular after that one part state was dissolved). Clan based societies tend to have difficulty transitioning to modern bureaucratic Western style states because loyalty to kin (especially in societies that have lots of cousin marriage) exceeds loyalty to the state.
Tribal governance There is quite a bit of variation in the governance and organization of small groups of ethnically united geographically isolated people into what are commonly called tribes of indigenous peoples all over the world from the Kalahari, to Australia, to the deep Amazon, to pre-contact North America and Meso-America. Often it has clan system similarities but is more varied something possible in part due to smaller scale.
Direct Democracy (e.g. early New England towns, Greek City-states, pirate ships, some colonial military units). Closely related would be the multi-focal organization of universities like Oxford and Cambridge in England into multiple independent but affiliated colleges.
Oligarchies. Most of the nominally democratic pre-18th century small representative democracies mostly in European city-states (Venice was particularly arcane and quirky in its political organization) eventually evolved into more or less hereditary oligarchies in which a small number of families passed political leadership posts from one generation to the next more or less without question, but no on family had control making it democratic within the leadership group. Pre-Democratic Poland which had a very large aristocracy relative to its population also had some similarities to this. Consider this example from Venice (also influenced by lottery ideas discussed below):
For more than five centuries (from 1268 to 1797) the procedure to
elect the doge (chief of state) did not change.
Choose 30 members of the Great Council by lot.
These 30 people are reduced by lot to 9.
These 9 people choose 40 other people.
These 40 are reduced by lot to 12.
These 12 people choose 25 other people.
These 25 people are reduced by lot to 9.
These 9 people choose 45 other people.
These 45 people are reduced by lot to 11.
These 11 people choose 41 other people.
These 41 people elect the doge.
Lottery leadership This is more theory than practice, although some Greek city-states and Venice may have come close. Bahai political theory (which was closer in geographic origins to the historic Venetian Republic than most people realized and not so remote temporally either) toys with the concept that the system should choose the leaders rather than just taking volunteers in order to keep power hungry psychopaths out of positions of power. The common law jury system evidences some of this notion as well.
Consensus based organization While no national state is (or to my knowledge ever has been) organized on the basis the Society of Friends (i.e. the Quakers) organizes its denomination on a consensus governance model with lots of lots of long meetings, and has been organizationally very effective relative to their numbers by doing so.
Self-governing subcultures. There are a number of cases where an ethnic community within a larger state establishments many institutions that would often be provided by the state for the community only. Roman Catholics in the U.S. created hospitals, K-12 schools and colleges as well as charities. Jews, being excluded from other institutions often had to do something similar (for example Jewish Community Centers, Jewish day schools, Jewish colleges, Jewish country clubs and often Jewish law firms). Asian immigrants to the U.S. especially in California established mutual aid societies that fulfilled some of these roles. Jati in India are separate communities of mutual support. Family law and inheritance is delegated to religious communities rather than being uniform in both India and Israel.
I'll add more if I think of them, but this is at least a start.
Anthropologists and sociologists have a model of stages of political complexity that must evolve in sequence to reach higher levels.
Anthropologists also link social and political structure differences to historical means of production. Many hunter-gatherer societies are organized around brother-sister households. Similarly a history of hoe farming tends to be associated with more matriarchy while plough farming tends to be associated with patriarchy. Polyandry tends to emerge in very economically marginal conditions where one man can't support a wife by himself, while polygyny tends to emerge in clan based patriarchal societies.
Also the idea of left wing and right wing form of dictating life is very much not universal. While virtually every political system has factions, plenty of them don't meaningfully fall on left wing v. right wing lines. This is obscured somewhat because a lot of the most familiar examples are in mostly ethnically homogeneous nation-states. But, in lots of political systems ethnicity or religion, rather than right wing v. left wing, are very important dimensions of political factions. Close to home is the example of the Quebec independence movement in Canada. The basic policies of Quebec independence movement supports aren't far out of the mainstream of Canadian political thought at all and are rather centrist overall. But, they are all about ethnic autonomy and identity. This is divide is true of probably a majority of former European colonies, especially in Africa, but also, for example, in Thailand or Singapore. And, even when you can superficially slot one faction as to the left or right of another, the terms really lose meaning at some point. Spain is another country where the ethic divide dimension is very important. Also Northern Ireland and India.
A left-right divide also has a very different meaning for someone advocating for land reform in Zambia where historically European plantation type farm owners controlled almost all of the arable land, or for a Maoist Revolutionary in India who is really a closer cousin to the anti-monarchists of 19th century Europe than to the 20th century Chinese political faction from which it gains its name, than for Democrats and Republicans in the U.S. or Labour and Tories in the U.K.