What is the structure of an oligarchy and what are some past or present countries with an oligarchy?

An oligarchy is "rule by the few". It does not specify how those few are defined, though traditional methods are by membership in some organization, birth into a specific set of families (or social class/caste), or wealth.

In modern usage, it generally EXCLUDES forms of government like a monarchy or dictatorship, in that an oligarchy generally uses some form of group decision-making. For the most part, the modern oligarchy is structured much like a modern republic or democracy, but "voting" is extremely limited.

Ancient Oligarchies would include the Holy Roman Empire (where Emperors were selected by a very limited number of Electors) and Japan under the Shoguns (where the Emperor was a figurehead and actual rulership was a competition between various Great Families).

Modern Oligarchies are harder to define, though I would personally classify many African nations as practical or de-facto oligarchies, in that while all citizens nominally can vote, the reality is that only a very limited number of families supply all the politicians, and control all the wealth and power in the country.