Oligarchy means the rule of the few, and those few are generally the people who are richer and more powerful than the others, what you might call the aristocrats or the nobles. These are not always men: just as monarchies have both kings and queens, women sometimes appear in councils of aristocrats, and even when they are not members, they are often there telling their husbands or their sons what to do. So oligarchies are generally bad for the poor, but they are pretty good for women, at least for rich women from powerful families.
Usually the way an oligarchy works is that there is a group of people who are in charge, somehow. Sometimes they may be elected, and sometimes they are born into their position, and at other times you might have to have a certain amount of money or land in order to be in the council. Then this group of people meets every so often – every week or every month – to decide important questions, and to appoint somebody to deal with things. Like they might decide that it should be illegal to steal, and then they would appoint one of the nobles to be a judge, and decide if people were guilty of stealing, and decide what to do with them if they were.
Oligarchies were not as common as monarchies in the ancient world. Right after the Dark Ages, most of the city-states in Greece were oligarchies, between about 1000 and about 500 BC. Athens was an oligarchy, and so were Thebes and Corinth. The Etruscans were also oligarchic. The Roman Republic, which started around 500 BC, was in some ways an oligarchy too.
In the Middle Ages, the cities of northern Italy – Genoa, Florence, Venice – were ruled by groups of rich people. Some were more like democracies, but others were more oligarchic. Outside of Europe, oligarchies have always been rare.