Athens had a superior navy.
It also had long walls protecting the city and its port.
This allowed Athens to withstand seige indefinitely as it could import food.
It could also send out its fleet to attack the home cities of its adversaries and collect funds from its empire to finance the war and its supply.
A Peloponnese because it is the peninsula forming the southern part of the mainland of Greece.
By sea.
Its Peloponnesian League opponents were superior on land. Athens' strategy was to defend the city walls and attack the opposition cities with its superior navy. This worked until persia financed a comparable navy for the Peloponnesian League.
Phase 1 - the Peloponnesian League invested Athens, which sent out amphibious expeditions targeting the Peloponnesian League cities. Phase 2 - a phony peace during which Athens tried to capture Syracuse in Sicily. Phase 3 - more direct hostilities during which Persia supplied the funds for the Peloponnesian League to mount a fleet which could match the Athenian one. Athens defeated at sea, invested and surrendered.
The Athenian empire was stronger at sea, the alliance led by Sparta was stronger on land.
Peloponnesian League invests Athens. Athens defends city walls, but raids Peloponnesian cities by sea. Persia funds Peloponnesian League navy. League reduces Athenian allies and defeats Athenian navy. Peloponnesian League invests Athens, sending its expatriate population home to help starvation force capitulation the sooner.
It was a standoff - Athens was dominant at sea and the Peloponnesians on land.
Sparta had the largest area after it conquered the southern Peloponnese, while Athens had limited land in Attica, until it created an empire for 50 years of the GREEK cities in the eastern Aegean Sea littoral.
Each had allies which augmented its power. Athens had an empire of about 180 city-states spread around the eastern Aegean Sea,. It had strong walls, which extended down to the sea, which enabled it to be resupplied during a siege. It also had a superior navy, which gave it the ability to threaten and raid opposing cities of the Peloponnesian League led by Sparta, and to ensure resupply of the city by sea from its empire. The Peloponnesian League had the strongest land forces, and was able to besiege Athens. By whittling don Athens' allies, it progressively reduced Athens' power and resupply, until finally after 27 years of fighting, the Athenian fleet was destroyed (Persia gave money to pay for a Peloponnesian fleet) and Athens was starved into submission.
Athenians mostly were very good sailors and they had the strongest navy in Greece. During the Peloponnesian War, they defeated Sparta in countless ocean battles.
When Athens used its navy to good effect during the 27-year Peloponnesian War, Sparta realised that it had to get the Peloponnesian league an equally strong nave to succeed. They took money from Persia to build a fleet which finally destroyed the Athenian navy and led to Athens' surrender.
Athens' empire was built on the Delian League which was formed to contain Persian invasion. Once peace was arranged with the Persian Empire, Athens kept those states under control and turned them into an empire of its own.
The Peloponnesian war was Between Athens and Sparta. Sparta was jealous of Athens and hated their idea of democracy. Sparta was strong on land, Athens at sea. But in the end, Sparta won and established an Oligarchy called the 30 tyrants. But 50 years later when Sparta was still weak from fighting Athens, a kingdom to the north called Macedonia, led by Alexander the Great, conquered Sparta.