Your class has studied four kinds of government. They were oligarchy, aristocracy, monarchy, and democracy. Monarchy and democracy are easy to distinguish from one another. Monarchy is rule by one person. In a monarchy, a king or a queen rules and that person has all the power. In Greece, there were only kings and no queens. In a democracy, the people rule. In Greece, this meant that the adult free males ruled. These free adult men got to assemble and vote on proposed laws. They were the ultimate source of authority.
Oligarchy and aristocracy are less distinct from one another. Both of these can be defined as rule of the few. However, there is a difference in how “the few” are selected. In an aristocracy, the few are selected on the basis of their family ties. People of a given few families have the right to be part of the few. In an oligarchy, the selection is generally on the basis of power or wealth. The people who get to rule are at least people who have, in some way, proven that they deserve to rule. Thus, these are similar, but not identical types of government.
Here is my answer for question 2:
The Peloponnesian War was fought between the Greek states of Sparta and Athens ans respective allies. The chief conflict was over the gradual encroachment of the Delian League into the whole of Greece, which while once noble following the Battle of Artemisium and the leadership of Pericles, had degenerated into authoritarianism as exemplified by the Melian Dialogue.
Sparta was no champion of democracy but conflict was inevitable due to the above violations of their sovereignty.