An oligarchy is any form of government where power is controlled by a small group of people. This can be beneficial in smaller populations, particularly if a group like village elders is in power, because wiser people tend to be less selfish. But in larger populations, oligarchies tend toward being tyrannical and make a good idea for evil sovereign powers on our world. Anything can be the basis for an oligarchy but is usually something like wealth, military power, status, family, higher education, or ability (like wizards). Owners of large corporations could qualify, especially in SF. There is sometimes a figurehead who appears to be the leader while the real power structure is unknown to the public. The military is used to maintain control and order.
Several types of oligarchies are discussed below.
An aristocracy is a form of government where a privileged class of people, supposedly the most qualified, rule. They might be the most famous and will usually be wealthy or otherwise influential. Belonging to the aristocracy requires inheriting the right or having it conferred upon one by the monarch. The people have few if any rights while the aristocrats have many and might even be above the law.
Historical examples include medieval nobility in Europe and ruling classes in India, Athens Greece, and Rome.
In a plutocracy, a small group of rich people are in control. The leaders may enrich themselves at the expense of the poor. And they do not make life better for everyone, just themselves, so we typically won’t see them creating social programs to aid the general population. In such a sovereign power, opulence will dominate areas where the rulers dwell while poverty might spread everywhere else.
Historical examples include merchant republics in Venice, Florence, and Genoa, and the Holy Roman Empire.
A military junta results from a military conquest of a country, the power now being held by multiple military leaders as a political group. The state is fundamentally authoritative, as you’d guess, and there are no elections. There’s also no constitution or laws adding legitimacy. Those in power typically emerge as leaders after combat, whether personal or military.
Past examples include Thailand, Burma/Myanmar, and Argentina.
Like a junta, a stratocracy is a military government, but this one has a constitution, laws and formal government, where every position is held by officers. If people can join the military, voluntarily or not, they become eligible to be part of this government. They therefore acquire the right to vote, for example, provided they are in good standing, such as having been honorably discharged. The rights of people are often limited, much the same way any military places limits on the conduct of its population. Since officers can be promoted on merit, this can be a meritocracy where only the most worthy advance.
Examples include Burma/Myanmar and the fictional Cardassian Union of Star Trek.
In a timocracy, only property owners can participate in government. No one else can hold office or vote, for example, and their rights are limited. Acquiring property might be difficult. It’s easy to imagine red tape and other barriers, or plots to rob an heir of property that would be inherited. Perhaps the government has recently eliminated the inheritance of property by individuals, which goes to itself instead of the heir.
We could also have a magocracy, where only people who have magical power can be members of government, hold office, or vote, and no one else has much in the way of rights. There could other versions of this, replacing wizards with one thing or another—vampireocracy, undeadocracy, or elfinocracy. The latter could be more simply a “raceocracy,” a term used regardless of the race in power, although it doesn’t sound as good, but you get the idea.
Other ideas, which are sometimes theoretical or just rarely seen, have been suggested, going all the way back to Plato. Each of them amounts to rule by a select group, whether it’s the strong, wise, technologically advanced, or whatever other criteria we invent. This is an opportunity to make things up. Another variant is rule by thieves, such as might be found in a corrupt city or pirate den. Rule by corporations or banks might be useful in futuristic SF. For more ideas, visit https://artofworldbuilding.com/government.